spectral analysis of output devices, from printing to...
TRANSCRIPT
SPECTRAL ANALYSIS OF OUTPUT DEVICES, FROM PRINTING TO PREDICTING
by
Behnam Bastani MSc. Computer Science, Simon Fraser University, 2005 BSc. Computing Science, Simon Fraser University, 2003
THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
In the School of Computing Science
© Behnam Bastani, 2009
SIMON FRASER UNIVERSITY
Summer, 2009
All rights reserved. This work may not be reproduced in whole or in part, by photocopy
or other means, without permission of the author.
ii
APPROVAL
Name: Behnam Bastani Degree: PhD Title of Thesis: Spectral Analysis of Output Devices, from printing
to predicting Examining Committee: Chair: Dr. Torsten Möller
Associate Professor of Computing Science
______________________________________
Dr. Brian Funt Senior Supervisor Professor of Computing Science
______________________________________
Dr. Tim Lee Supervisor Computer Scientist, Cancer Control Research, BC Cancer Agency
Assistant Professor, Dermatology and Skin Science, UBC
Adjunct Professor, Computing Science, SFU
______________________________________
Dr. Ghassan Hamarneh Internal Examiner Assistant Professor of Computing Science
______________________________________
Dr. Raja Bala External Examiner Principal Color Scientist at Xerox
Date Defended/Approved: June 29, 2009
Last revision: Spring 09
Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users.
The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the “Institutional Repository” link of the SFU Library website <www.lib.sfu.ca> at: <http://ir.lib.sfu.ca/handle/1892/112>) and, without changing the content, to translate the thesis/project or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work.
The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies.
It is understood that copying or publication of this work for financial gain shall not be allowed without the author’s written permission.
Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence.
While licensing SFU to permit the above uses, the author retains copyright in the thesis, project or extended essays, including the right to change the work for subsequent purposes, including editing and publishing the work in whole or in part, and licensing other parties, as the author may desire.
The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive.
Simon Fraser University Library Burnaby, BC, Canada
iii
ABSTRACT
The focus of this thesis is to develop and introduce algorithms that extend
traditional colour reproduction from three dimensions to higher dimensions in
order to minimize metamerism. The thesis introduces models that can accurately
predict interactions between the primaries for non-linear output devices in
spectral colour space. Experiments were designed and performed to aid in
understanding how optimized the spectral characteristics of existing printer inks
and display primaries are, and how the inks and primaries should be designed so
that the accuracy of the reproduction is optimized.
The time and space computational complexity of the reproduction
algorithms grows exponentially with the number of input dimensions. The
algorithms for finding the best combinations of inks or primaries matching a given
input reflectance become more challenging when the inks interact with each
other non-linearly, as is usually the case in printers. A number of different
methods are introduced in this thesis to handle gamut mapping and the colour
reproduction process in higher dimensions. An ink-separation algorithm is
introduced to find the ink combination yielding a chosen gamut-mapped spectral
reflectance. Experiments with real inks for spectral colour reproduction were
performed to compare the results of the reproduction against trichromatic colour
reproduction on a 9-ink printer system. Finally, a new application of reflectance
analysis in higher dimensions is introduced.
iv
Keywords: Spectral Reproduction, Spectral Printing, Modelling, Gamut Size, Bronzing, Gamut Mapping Subject Terms:
v
DEDICATION
I would like to dedicate this work to my father and my mother, Bijan and
Mahnaz and my lovely wife Parinaz.
vi
ACKNOWLEDGEMENTS
I would like to thank Dr. Jeffrey Dicarlo, Dr. Jan Morovic and Dr. Tim Lee
for assisting me through this research. I also thank Dave Hunt, Rick Becker and
Amy Van Liew for their financial and management support at Hewlett-Packard
Company.
Last, but not the least, I would like to express my heartfelt gratitude to my
senior supervisor, Dr. Brian Funt, for his encouragement and for patiently
supervising and guiding me for the past four years.
vii
TABLE OF CONTENTS
Approval .............................................................................................................. ii Abstract .............................................................................................................. iii Dedication ........................................................................................................... v
Acknowledgements ........................................................................................... vi Table of Contents ............................................................................................. vii List of Figures ..................................................................................................... x
List of Tables .................................................................................................. xvii Glossary ........................................................................................................... xix
Chapter 1: Spectral Reproduction .................................................................... 1
Introduction ....................................................................................................... 1 Survey and Proposal Layout ............................................................................. 2
Chapter 2: Mathematical Preliminaries ............................................................. 5
Thin Plate Spline Interpolation ....................................................................... 5 Principal Component Analysis ....................................................................... 6 Least Square and Non-Negative Least Square Method ................................. 7 ISOMAP and Multi-Dimensional Scaling ........................................................ 7
Chapter 3: Printer Modelling ............................................................................ 11
Light and Media Interaction and the Dot Gain Phenomenon ........................... 11 Modelling Ink and Medium interaction ............................................................. 14
Linear Model (Murray Davies) [15], [9], [17] ................................................. 14 Kubelka Munk Model [15], [16]..................................................................... 15 Neugebauer Model ([18], [20], [21], [15]) ..................................................... 17
Challenges of Ink Modelling and Printer Characterization ............................... 20 Smarter Sampling: Uniform Sampling in Perceptual Colour Space ............. 21 Physical Constraint: Ink Limiting .................................................................. 22 Handling the Missing Points (Neighbours) ................................................... 25 Smarter Transformation and More Advanced Interpolation Method............. 27
Data Collection ................................................................................................ 28 Implementation ................................................................................................ 29 Results 30
Results: Modified YNCN (Smarter Sampling and Dealing with Missing Neighbours) .................................................................................. 30
Result: Manifold Based Printer Model .......................................................... 33
viii
Chapter 4: Number of primaries ...................................................................... 38
Calculating Data Set Complexity ..................................................................... 38 Rotated PCA Basis .......................................................................................... 39
PCA Eigenvector without Sample Mean ...................................................... 41 Multi-Peak Primaries [30] ................................................................................ 44 Analysis of Primary Characteristics ................................................................. 47
Primary Selection ......................................................................................... 47 Device Characteristics ................................................................................. 51
Evaluation Method ........................................................................................... 52 Hierarchical Search (HS) Gamut Mapping Algorithm ................................... 54 Optimizing the HS Parameters..................................................................... 55 Evaluating Metamerism ............................................................................... 57 Scene Data Base and K Means ................................................................... 58
Results 60 Primary Overlap ........................................................................................... 60 Primary Interaction Model ............................................................................ 64 Number of Primaries .................................................................................... 68
Chapter 5: Spectral Gamut Mapping and Spectral Ink Separation ............... 72
Spectral Reproduction Based on Interim Colour Space .................................. 73 Spectral Ink Separation Based on Inverting a Printer Model ........................... 77 Proposed Method I: Spectral Ink Separation using Non-Negative Least
Squares ............................................................................................ 77 Preserving Colour for a Desired Illumination ................................................ 79 Evaluation of Gamut Convexity .................................................................... 81 Experiment ................................................................................................... 82 Results 83
Proposed Method II: Geodesic Based Ink Separation for Spectral Printing ............................................................................................. 88
Use of Thin Plate Spline Interpolation in Spectral Reproduction .................. 89 Geodesic Interpolation and Ink Separation .................................................. 89 Spectral Gamut Mapping based on Manifold Projection .............................. 90 Evaluation Method ....................................................................................... 93 Time and Space Complexity ........................................................................ 94 Experiments ................................................................................................. 94 Printer Spectral Gamut Intrinsic Dimensionality ........................................... 95 Results 98 Spectral Gamut Mapping Evaluation ............................................................ 99
Conclusion .................................................................................................... 104
Chapter 6: Evaluation of Spectral Colour Reproduction............................. 105
Introduction ................................................................................................... 105 Target Samples ............................................................................................. 105 Experiment Setup .......................................................................................... 108 Implementation Details .................................................................................. 110 Results 111
ix
Chapter 7: Spectral Analysis of Bronzing .................................................... 115
Introduction ................................................................................................... 115 Data Measurement ........................................................................................ 117 Bronzing Evaluation based on Spectral Reflectance Characteristics ............ 117
Results 121 Modeling Bronze ........................................................................................... 123
Predicting Reflectance under Different Viewing Angles ............................. 123 Modelling Bronze for Different Ink Densities and Phase Angles ................ 125 What can be learned from this model? ...................................................... 126 Experiment Setup ...................................................................................... 127 Results 128
Chapter 8: Summary ...................................................................................... 130
Detailed Contributions ................................................................................... 133
References ...................................................................................................... 136
x
LIST OF FIGURES
Figure 1: Swiss Roll representation in 3 Dimensions [86] ..................................... 9
Figure 2: Un-folded Swiss Roll data into 2 dimensions using ISOMAP. [86] ...... 10
Figure 3: Primary Interactions between Light and Medium (paper) are scattering and absorption ([75]). .......................................................... 11
Figure 4: Physical dot gain causes a drop of ink to cover a larger area that expected from a perfect linear production. .......................................... 13
Figure 5: Optical dot gain occurs because the halftone dots are printed on a scattering substrate. a0 shows the original drop size, ∆aphy is the physical dot gain and ∆aopt is the optical dot gain. ......................... 13
Figure 6: Dot gain curve. Maximum dot gain occurs around 50% of area coverage, where there is enough space left for physical dot gain and optical dot gain without interacting with neighbouring dots. .......... 14
Figure 7: Kubelka Munk absorption and scattering theory. ................................. 15
Figure 8: Valid Patches for 2 inks with U=1. ....................................................... 23
Figure 9: Valid Patch Space for ½ < U < 1. ........................................................ 24
Figure 10: Correlation of missing neighbours and error for Extrap=1 method, R2=.38. .................................................................................. 32
Figure 11: Residual error of spectral recovery as a function of reflectance dimensions. The data is for spectral measurement and spectral measurement after a logarithmic transformation. ............................... 34
Figure 12: Residual error for spectral recovery of the 8-ink printer spectral gamut after ISOMAP transformation. ISOMAP (LOG) shows when a Yule-Nielson of value 2 is applied to the ISOMAP transformed gamut data. ..................................................................... 35
Figure 13: Performance of Geodesic (ISOMAP) and Linear based modelling compared against the YNCN model. The vertical axis represents the average error calculated as the Root Mean Square difference between predicted reflectance and measured values. ................................................................................................. 37
Figure 14: The six eigenvectors obtained from the still life painting by Di-Yuan [27]. ............................................................................................ 40
xi
Figure 15: The six acrylic-paints used for generating the sample population. The vertical axis shows the K/S factor of Kubelka Munk theory, a representation of reflectance (absorption/scattering). ........................................................................ 42
Figure 17: The estimated colorants (solid lines) and the original colorants (marked as start) used in the painting data base. ................................ 44
Figure 18: The effect of increasing the number of sensors with 12-bit quantization and 1% shot noise. .......................................................... 46
Figure 19: 3 Square wave synthetic ink reflectances covering 380 to 730 nm equally. .......................................................................................... 48
Figure 20: Two primaries with non-smooth tailed endings. The reflectances of these primaries have a gradual transition between absorptive and non-absorptive regions. ................................ 49
Figure 21: Two primaries with smooth tailed endings. The reflectances of these primaries have a smooth gradual transition between absorptive and non-absorptive regions. ............................................... 49
Figure 22: Reflectance of real inks measured. The curves show the smooth and long tails that are common for real reflectances. ............. 50
Figure 23: Square-wave reflectance functions with 0% overlap. ........................ 50
Figure 24: Square-wave reflectance functions with 20% overlap. ...................... 51
Figure 25: Mean Root Mean Square residual error as the number of PCA bases increases. .................................................................................. 59
Figure 26: Mean DeltaE94 error between reconstructed reflectance and the database reflectance when different numbers of PCA bases used. DetaE94 is calculated as average detalE under 11 different illuminations provided from the SFU database. ................................... 60
Figure 27: Spectral Gamut coverage of square wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on non-linear (TK) model. .......................................................................................... 61
Figure 28: Spectral Gamut coverage of square wave ink given 3, 6, 9 and 12 inks evaluated as DeltaE94 colour difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The error is shown as average variation under 11 different illuminations. The printer model is based on non-linear (TK) model. ......................................................................... 61
Figure 29: Input reflectance and closest match using 6 ink square waves with 0 and 20 % overlap. The printer model is based on non-linear (TK) model. ................................................................................ 62
xii
Figure 30: Spectral Gamut coverage of square wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on linear (LP) model. .................................................................................................. 62
Figure 31: Spectral Gamut coverage of trapezoidal wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square (RMS) difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on a non-linear (TK) model. ......................................................................... 63
Figure 32: Input reflectance and closest match using 6 ink square waves with 0 and 20 % overlap. The printer model is based on a non-linear (TK) model. ................................................................................ 63
Figure 33: Spectral Gamut coverage of sine wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square (RMS) difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on a non-linear (TK) model. ......................................................................... 64
Figure 34: Spectral Gamut coverage of sine wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square (RMS) difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on a linear (LP) model. ................................................................................ 64
Figure 35: Spectral match of a scene reflectance using a 3-ink system with square wave inks of 20% overlap. The reproduction is shown for linear and non-linear systems. For linear system the reproduction error has RMS value of .11 and non-linear system has .065 ............................................................................................... 65
Figure 36: Performance of square wave primary evaluated as RMS of match between scene reflectance and the closest match on the system gamut. ..................................................................................... 66
Figure 37: Performance of square wave primary evaluated as RMS of match between scene reflectance and the closest match on the system gamut. ..................................................................................... 67
Figure 38: Performance of square wave primary evaluated as average RMS of match between scene reflectance and the closest match on the system gamut. .......................................................................... 67
Figure 39: Performance of real ink reflectances as average RMS of match between scene reflectance and the closest match on the system gamut (both linear and non-linear models are evaluated). .................. 68
Figure 40: Spectral Gamut coverage of sine wave ink at 20% overlap (the better overlap amount) for 3, 6, 9 and 12 inks evaluated based
xiii
on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis. .............................. 69
Figure 41: Spectral Gamut coverage of square wave ink at 0% overlap (the better overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis. ................... 70
Figure 42: Spectral Gamut coverage of sine wave ink at 40% overlap (the undesired overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis. ................... 70
Figure 43: Spectral Gamut coverage of square wave ink at 20% overlap (the undesired overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis. ................... 71
Figure 44: Spectral Gamut coverage of square wave ink at 40% overlap for a linear system. 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis. ........................................... 71
Figure 45: Gamut coverage calculated using the non-negative least square technique. The Y axis shows Root Mean Square (RMS) variation between mapped reflectance and the input scene reflectance. SQ0 represents square wave inks with 0% overlap, Sine10 represents sine wave inks with 10% overlap, and Realistic represents real ink reflectance measurements. .................... 84
Figure 46: Variation between convex gamut mapping and concave gamut mapping. The concave gamut mapping is based on the HS iterative search algorithm and the convex gamut mapping is based on the non-negative least square algorithm. The variation is calculated as average RMS (Root Mean Square) difference between the two spectral reflectances calculated using two different mapping methods. ................................................................. 85
Figure 47: Variation between convex gamut mapping and concave gamut mapping for primaries with smooth variations (sine wave). The concave gamut mapping is based on the HS iterative search algorithm and the convex gamut mapping is based on the non-negative least square algorithm. The variation is calculated as the average RMS (Root Mean Square) difference between the two spectral reflectances calculated using the two different mapping methods. ............................................................................... 86
Figure 48: Variation between convex gamut mapping and concave gamut mapping for primaries with smooth variations (sine wave). The concave gamut mapping is based on the HS iterative search algorithm and the convex gamut mapping is based on the non-
xiv
negative least square algorithm. The variation is calculated as average DeltaE94 difference between the two spectral reflectances calculated using two different mapping methods. ............ 87
Figure 49: Reflectance characteristics of HS and NNLSQ gamut mapping for 6 and 12 ink systems. The inks are sine wave inks with 20% overlap. ................................................................................................ 88
Figure 50: Example of the gamut mapping algorithm in Euclidean Space. Blue lines (Swiss roll) represent a device gamut and the green point represents an out of gamut point. ............................................... 92
Figure 51: Gamut mapping method using ISOMAP where the projections are applied in a lower dimensional space. ........................................... 93
Figure 52: PCA residual variance for the realistic printer gamut space. The plot shows that the dimensionality of the 7-ink printer is around 5 dimensions. ......................................................................................... 96
Figure 53: ISOMAP residual for the realistic gamut. The data shows that the underlying dimensionality of the gamut is around 3. ...................... 97
Figure 54: PCA residual variance for the synthetic printer gamut space. The scores show that the dimensionality of the 6-ink printer is around 6 or 7 dimensions. ................................................................... 97
Figure 55: ISOMAP residual for the synthetic gamut. The data shows that the underlying dimensionality of the gamut is around 3 to 4. ............... 98
Figure 56: Ink separation methods evaluated in ∆E94 under 11 different illuminations. The data above shows average ∆E94 for the 11 illuminations ......................................................................................... 99
Figure 57: Comparison between ISOMAP-based gamut mapping and HS gamut mapping for realistic and synthetic ink reflectances. The variation is calculated as the average RMS difference between the two spectral reflectances calculated using the two different mapping methods. ............................................................................. 101
Figure 58: Comparison between ISOMAP-based gamut mapping and HS gamut mapping for realistic and synthetic ink reflectances. The variation is calculated as average DeltaE94 difference between the two spectral reflectances calculated using the two different mapping methods. ............................................................................. 101
Figure 59: Comparison between accuracy of ISOMAP-based gamut mapping and NNLSQ gamut mapping measured in mean RMS. Accuracy is defined as variation between how the gamut mapping performs compared to HS gamut mapping ......................... 102
Figure 60: Comparison between accuracy of ISOMAP-based gamut mapping and NNLSQ gamut mapping measured in mean DeltaE94 for square wave ink with 0% overlap. Accuracy is
xv
defined as variation between how the gamut mapping performs compared to HS gamut mapping ....................................................... 103
Figure 61: Comparison between accuracy of ISOMAP-based gamut mapping and NNLSQ gamut mapping measured in mean DeltaE94 for real inks. Accuracy is defined as variation between how the gamut mapping performs compared to HS gamut mapping ............................................................................................. 103
Figure 62: The 3 colour tiles used from the MacBeth Colour Checker for spectral colour reproduction testing. The colour tiles used were the 5th, 8th and 13th colour tiles as indicated by the red mark. ............ 106
Figure 63: Reflectance characteristics of the 3 colour patches selected from the MacBeth Colour checker as target reflectances for reproduction....................................................................................... 107
Figure 64: Reflectance characteristics of the 3 yellow paint samples selected as targets. ........................................................................... 108
Figure 65: Reflectance characteristics of the 3 different magenta inks used for the experiment. ............................................................................. 109
Figure 66: Reflectance characteristics of the 2 different red inks used for the experiment. .................................................................................. 109
Figure 67 Reflectance characteristics of the 2 different yellow inks used for the experiment. .................................................................................. 110
Figure 68: Accuracy of reproduction of patch number 5 under 4 different illuminations using trichromatic matching versus spectral colour reproduction. [84] ............................................................................... 112
Figure 69: Accuracy of reproduction of patch number 13 under 4 different illuminations using trichromatic matching versus spectral colour reproduction....................................................................................... 112
Figure 70: Accuracy of reproduction of patch number 13 under 4 different illuminations using trichromatic matching versus spectral colour reproduction....................................................................................... 113
Figure 71: Accuracy of reproduction of the yellow patch S_Y2 under 4 different illuminations using trichromatic matching versus spectral colour reproduction. The trichromatic matching was done under Incandescent light A illumination. ................................... 114
Figure 72: Spectral Reflectance Variation of black ink (K1) as viewing angle changes (keeping incident angle constant). ............................. 118
Figure 73: Spectral reflectance variation of the black ink (K1) under different viewing angles normalized by reflectance of the white paper under the same angles. ........................................................... 119
xvi
Figure 74 Reflectance of K1 ink measured at two different viewing angles. The data is plotted against the difference between the viewing and incident angles. ........................................................................... 120
Figure 75: Maximum reflectance variation of an ink under two different densities when considered under a set of viewing angles (keeping incident angle constant). The selected set of viewing angles are+- 10 degrees of incident angle. ........................................ 121
Figure 76: Spectral characteristics of the 5 inks used in the study. K1, K2 and K3 represent 3 different types of black ink tested. C represents cyan ink and M represents magenta ink. ......................... 122
Figure 77: Bronzing level calculated based on the proposed bronzing metric. K1, K2 and K3 represent 3 different types of black ink tested. C represents cyan ink and M represents magenta ink. .......... 123
Figure 78: Variation of the black ink (K1) reflectance at 480 nm wavelength under different viewing angles. The incident angle was kept constant at 20 degrees. ...................................................... 124
Figure 79: Reflectance variation of 3 different ink densities at 480 nm wavelength. ....................................................................................... 126
Figure 80: Spectral characteristics of the 2 black inks used in this study. The measurements were collected using an eye-1 spectrophotometer at 2 degrees observer angle. .............................. 128
Figure 81: Performance of the bronzing model compared to real measurements. The vertical axis represents the bronzing metric that was proposed in the previous section. D1 to D5 show the 5 different densities used for each ink density. D1 is lightest ink density and D5 is the highest ink density. K1 and K2 are the 2 black inks used. ................................................................................. 129
xvii
LIST OF TABLES
Table 1: YNCN Performance for an 8-ink printer with missing Neighbours.
Mean, Max and Std represent average, maximum and standard deviation of the error respectively [68] ................................................. 31
Table 2: Correlation between missing score (percentage of missing neighbours) and error in the model (∆E94) .......................................... 32
Table 3: Performance of Geodesic (ISOMAP) and Linear based modelling of the 8-ink printer system using TPS (Thin Plate Spline) interpolation. ISOMAP (LOG) and Spectral (LOG) represents TPS interpolation applied to each space after they have gone through a logarithmic transformation (which is similar to having the Yule-Nielson factor equal to 2). ..................................................... 36
Table 4: Colorimetric accuracy of the six estimated colorants for the painting database ................................................................................ 43
Table 5: Performance of the HS search for 3, 6, 9 and 12 ink printer systems with the LP Printer model. The performance was compared against the linear projection method that returns the true answer. Square wave inks were used in the experiment. ............. 57
Table 6: The 11 different illuminations used in measuring the colour variation of two similar reflectance spectra. This data is from Computational Vision Lab at Simon Fraser University [65]. ................. 57
Table 7: LABpqr performance of an 8-ink printer system in reconstructing 5 different data sets. Leeds database represents scene reflectances provided by Leeds University [82]. Munsell database includes reflectances for a set of prints and paintings measured by Leeds university. The SFU dataset is provided by Computational Vision Lab at Simon Fraser University [65]. The last two databases include the MacBeth Colour Chart and some pigmented colours used by artists. ...................................................... 76
Table 8: Ink separation evaluation based on geodesic location and linear space locations. The errors reported are the minimum, mean, and max ∆E94 that occur under the 11 different illuminations, and the RMS difference between the spectra. ............................................ 99
Table 9: Average Colour variation (inconsistency) of each patch under 11 different illuminations. Mean DeltaE94 column represents average colour variation from target patch ........................................ 107
xviii
Table 10: Colour variation (inconsistency) of the 2 yellow paint patches under 11 different illuminations. ......................................................... 108
Table 11: 4 Light sources available in the viewing light booth .......................... 110
xix
GLOSSARY
∆E94 DeltaE is a measurement of distance in CIELAB94 space. A value of one indicates a just noticeable difference in colour.
∆ECIECAM0
2 Changes in CIECAM02 colour space. A value of one does not necessarily represent one noticeable change.
CMYK cyan, magenta, yellow, black. In some cases, K is used to represent the grey axis
CIELAB94 International Commission On Illumination L a* b* colour space. This is a perceptually uniform colour space, where a unit of distance anywhere in the space is intended to represent the same amount of perceptual difference
LUT Look Up Table
SVR Support Vector Regression
XYZ Used to refer to the CIEXYZ tristimulus space, where X and Z represent chroma and Y represents luminance.
TPS Thin Plate Spline Interpolation
1
CHAPTER 1: SPECTRAL REPRODUCTION
Introduction
In comparison to standard colour reproduction, spectral reproduction aims
to reproduce a given reflectance spectrum rather than produce a metameric
reflectance spectrum that simply matches a given colour. This approach attempts
to reduce the problem that can arise in metameric colour printing or display,
which is that the reproduced colour may match under one illuminant, but not
match well under some other illuminant. Clearly, if the reproduced output
reflectance matches the input reflectance, the reproduced colour will match the
input colour under all illuminants.
Spectral reproduction has application in many fields. In fine art
reproduction, it is important to reproduce a painting in spectral space as closely
as possible to minimize metamerism. Bastani et al. [60] ) showed that if the
reflectances of a given set are optimized, fine art reproduction accuracy can be
improved by as much as 30%.
Another application is in high-end photography where professional
photographers would like to capture their subjects as accurately as possible and
in some cases reproduce them accurately. Soft proofing is another area where
the end-user requires the displayed image to look as a close as possible to the
real target under a range of illuminations.
2
Spectral colour reproduction becomes feasible when there is more than
one possible primary combination to match a colour. In this paper, primaries refer
to inks in a printer, phosphors or colour channels in a monitor or filters in a
projector. To have a more accurate colour reproduction, a larger number of
primaries are needed that are fairly independent from each other in their
wavelength coverage. However, the larger the number of primaries, the higher
the computational complexity of printing or displaying algorithms in terms of both
time and space. In particular, standard gamut-mapping algorithms map colours
within a 3-dimensional space and their computational complexity increases
rapidly with dimension. For example, a gamut-mapping algorithm that relies on
the computation of the convex hull of the measured gamut will not work for
spectral data with, say, 11 dimensions since computing a d-dimensional convex
hull of n points requires order O(nfloor(d/2)+1) operations.
Survey and Proposal Layout
The first complication in spectral reproduction is modelling the output of a
device accurately in spectral space. The modelling becomes more complex when
the primaries interact non-linearly such as in a printing environment. A naïve
approach is to measure all primary combinations and store them in a database.
This means if each ink dimension is sampled N times and there are D inks in the
system, then there are ND primary combinations to measure. This means that in
order to measure ramps of 255 samples along each primary axis with 8 primaries
in the system, there are 1.8x1019 patches to measure and store, which is not
practical.
3
The non-linear interaction of the inks with each other and with the paper
(medium) makes modelling the printer output an important component of the
research. Since the complexity of the proposed algorithms grows exponentially
with the number of inks, spectral modelling of an N-ink printer system is a
particularly significant topic. Two possible opportunities are discussed in
thisthesis: The first is to reduce the necessary number of patches to measure by
considering constraints such as ink limiting (amount of the ink that a medium can
accept). The second path considers lowering the complexity of the printer gamut
space before applying any interpolation method.
Spectral characteristics of the primaries and their interaction types,
whether linear or non-linear, have the largest effect on an output device gamut
size and the spectral reproduction accuracy of the devices. For instance, the
optimum selection of a set of inks for a printer may have less overlap or be more
spectrally independent of each other. The second part of this report presents
methods for optimizing the spectral characteristics of the primaries to improve the
spectral reproduction of a given system. It also presents how the effect on the
output device gamut and spectral gamut space of a variation in a primary
reflectance characteristic can be evaluated.
Given a set of ink reflectances and a device output model, the next
challenge is gamut mapping in spectral space. The traditional gamut mapping
algorithms work in three dimensions (CIELAB or CIEXYZ space) and they
typically assume that the gamut shape is convex. The complexity of calculating
the convex hull is another challenge that grows exponentially with the
4
dimensionality of the data. Two different gamut mapping approaches are
considered in this paper.
To evaluate how the spectral colour reproduction can help professional
users, an experiment is conducted to understand whether the proposed
algorithms are feasible in real life, given the available set of inks and printer
technology.
The last part of this study proposes other applications of spectral colour
analysis beside spectral colour reproduction.
5
CHAPTER 2: MATHEMATICAL PRELIMINARIES
In this section we introduce some mathematical concepts and notation
that will be used throughout the thesis.
Thin Plate Spline Interpolation
Thin Plate Spline, or TPS, is an interpolation method that finds a
"minimally bended" smooth surface that passes through most input data points.
The name "thin plate spline" came from the observation of bending a thin sheet
of metal. In the physical setting, the bending of the plate is known to be
orthogonal to the original plane structure. In order to apply this idea to the
problem of coordinate transformation in a 2 dimensional space, the lifting of the
plate can be interpreted as a displacement of the x or y coordinates within the
plane. This means, two thin plate spline functions (basis) are needed to specify a
two-dimensional coordinate transformation. The basis functions for TPS are the
Radial basis functions [13].
A radial basis function (RBF) is a function that its output value depends on
the Euclidean distance from the neighbouring input points, referred to as center
points. For TPS, a smoothness constraint is added to the fitting, where the
smoothness constraint minimizes the derivative variation between the output
points.
6
Xiong et al. extended the TPS model to N-Dimensions and applied it to
illumination estimation successfully [61].
For spectral printer modeling, TPS is used to find a set of continuous
function that each maps between the set of inks and oneof the output
dimensions. For instance if the output spectral reflectance of an 8-ink printer is
measured from 380nm to 730nm with a 10 nm sampling, TPS is used to create
36 separate functions mapping from the 8 input dimensions to each reflectance
wavelength 380nm, 390nm, …. to 730nm, separately.
Principal Component Analysis
Principal Component Analysis (PCA) [10] is a mathematical technique to
discover correlation behaviour between a set of variables. The method uses the
correlation information to break down the set of variables into subsets that are
relatively independent of one another. Variables in each subset are correlated
with one another and are largely independent of other subsets.
PCA is based on the orthogonal linear transformation of variables into a
new coordinate system such that in this new coordinate system, the largest
variance by any projection of the data falls on the first coordinate. Similarly, the
second largest variance is on the second coordinate and so on.
PCA is widely used in many fields including signal processing, statistics
and computer vision.
7
Least Square and Non-Negative Least Square Method
Least Squares regression, [11], is a mathematical method to solve
systems that have more equations than unknowns. Least Squares regression
finds the best-fitting curve to a given set of points by minimizing the sum of the
squares of the offsets (residuals).
Non-negative least squares finds the best fitting curve using the Least
Squares method with the additional constraint that all the coefficients of the fitting
should be non-negative. In this study the non-negative least square
implementation from Matlab is used [92].
ISOMAP and Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a classical technique for mapping the
input data to a lower dimensional space, subject to the constraint that pair-wise
distances between data points are preserved as much as possible. The latter is
accomplished via minimization of a cost function. The classical technique uses
the Euclidean distance metric.
ISOMAP or complete isometric feature mapping is a variant of MDS that
replaces the Euclidean distance with the geodesic distance.. For certain types of
data, ISOMAP can be more effective in uncovering the underlying structure and
dimensionality. Geodesic distance can be computed with a number of
techniques, a common one being Dijkstra’s algorithm. Error! Reference source
not found.[62] [88].
8
Dijkstra's algorithm finds the shortest path between the vertices in a graph
with nonnegative edge cost. This algorithm will be used to compute the geodesic
distance between a pair of points (detailed below). The time complexity of the
algorithm is O(N2), where N is the number of nodes (vertices) in a graph.
Specifically, the geodesic distances represent the shortest paths along the
curved surface of the manifold. This can be approximated by a sequence of short
steps or “hops” between neighbouring sample points. Since the manifold is not
known ahead of time, some heuristic should be used to define the neighbours of
a given point. In this thesis, the neighbours of a given point are defined as those
points whose path length to the given point are smaller than a threshold. The
threshold may vary based on how the points in a database are placed relative to
each other. If the threshold selected is too large then almost all the points are
selected as neighbours of each other and the ISOMAP technique may not be
able to map the space into an optimal low-dimensional space. On the other hand,
too small a threshold can cause a break in the space with the result that ISOMAP
may map the input space into multiple low-dimensional spaces.
In this study, a search is done to find a threshold that optimizes the
dimensionality reduction while stillincluding all the measurements points from the
printer gamut in a single space.
After selecting the neighbourhood threshold, ISOMAP then applies MDS
to the geodesic, rather than straight line, distances to find a low-dimensional
mapping that preserves these pair-wise distances.
9
The Swiss Roll is a common example used to visualize the difference
between geodesic and Euclidean distances. Figure 1 shows a Swiss Roll in 3
dimensions where points A and B have a small Euclidean distance (dotted line).
Figure 2 shows the Swiss Roll after the ISOMAP transformation into 2
dimensions. The figure shows that the two points have a large Geodesic distance
despite a small Euclidean distance relative to the other neighbouring points.
Figure 1: Swiss Roll representation in 3 Dimensions [86]
10
Figure 2: Un-folded Swiss Roll data into 2 dimensions using ISOMAP. [86]
11
CHAPTER 3: PRINTER MODELLING
As discussed earlier, modelling a device’s output accurately in spectral
space is essential to enabling spectral reproduction. In electronic displays, since
the light is added quite linearly, simple linear approaches are used to model the
displays’ outputs [1][2][3][4]. The situation is quite different in a printer where inks
behave in a non-linear fashion.
In this section, some basic background on printing systems and some well
known printer models are presented. In addition, two possible methods to
simplify and improve the accuracy of printer modelling are proposed.
Light and Media Interaction and the Dot Gain Phenomenon
When light hits a surface (medium), a percentage of the light gets
reflected and the rest is absorbed. These two percentages are referred to as the
scattering and absorption coefficients. Figure 3 illustrates this effect.
Figure 3: Primary Interactions between Light and Medium (paper) are scattering and absorption ([75]).
12
The reflectance behaviour for a material is different. For a fluorescent
material , the material (molecules) absorbs a photon (UV) and then emits a
photon (visible) of lower energy (longer wavelength). The effect of fluorescent
materials was not considered in the research reported here.
In the printing environment, when a drop of ink is put on the medium, there
are several physical and chemical interactions that happen which add to the
complexity of the system. The most common way of reproducing images on a
print is by half toning. Half toning produces different levels of grey or colours by
small dots with maximum colour density but with varying local fractional area
coverage, printed on a white substrate [12]. Whenever such a reproduction is
used, an effect that is referred to as dot gain happens, which makes the actual
image appear darker than what would have been expected from a perfect
reproduction. There are two parts to dot gain: physical dot gain and optical dot
gain. Physical dot gain occurs because the dots gain in physical size due to
imperfections in the image transfer from the original to the print (Figure 4). A
typical reason for physical dot gain is ink smearing and spreading in the printing
process which is normally referred to as ink and media interaction.
13
Figure 4: Physical dot gain causes a drop of ink to cover a larger area that expected from a perfect linear production.
Optical dot gain is the effect of a dot appearing larger than its actual size.
Optical dot gain occurs because the half tone dots are printed on a scattering
substrate (medium in printing systems) (Figure 5). The spread of light in the
medium causes a shadow around the rim of the dots which, in turn, causes the
dots to appear larger, represented as ∆aopt in Figure 5.
Figure 5: Optical dot gain occurs because the halftone dots are printed on a scattering substrate. a0 shows the original drop size, ∆aphy is the physical dot gain and ∆aopt is the optical dot gain.
Figure 4 shows the dot gain curve relative to the ink density. It shows that
the maximum dot gain occurs around 50% ink density, where there is enough
14
space left for physical dot gain and optical dot gain without interacting with
neighbouring dots.
Figure 6: Dot gain curve. Maximum dot gain occurs around 50% of area coverage, where there is enough space left for physical dot gain and optical dot gain without interacting with neighbouring dots.
Modelling Ink and Medium interaction
In this section, performance of the three best known approaches for
modelling printer output are summarized and compared.
Linear Model (Murray Davies) [15], [9], [17]
The Murray-Davies model is based on the assumption that the reflectance
of a half tone cell adds up linearly. This model estimates a single ink reflectance
on a medium using the following linear equation:
(1)
paperRaaRR ,%100, )1( λλλ −+=
where a is a function of dot area coverage and Rλ,100% is the spectral
reflectance of ink at 100% dot area coverage. Rλ,paper is the reflectance of the
white paper.
15
The Murray-Davies algorithm depends on the assumption of linear
interaction between ink dot area coverage and the medium. As discussed earlier,
because of some physical (dot gain) and chemical interaction between ink and
medium, this assumption does not always hold.
Kubelka Munk Model [15], [16]
The Kubelka Munk model is the most popular method for modelling printer
output. Kubelka Munk has a relatively simple equation. Its two parameters (K and
S) represent reflectance and transmission from a surface. Kubelka and Munk
examined the reflectance of a material having a thin layer of colorant in contact
with the opaque surface of the material [14], [15]. Kubelka-Munk theory is based
on the assumption that a colorant can be broken into a large number of thin
layers with equal optical properties (Figure 7). Figure 7 shows a colorant of
thickness X and two diffuse light fluxes I and J. The fluxes represent the overall
light that each layer receives or passes through each layer. The idea is that once
a colorant ink is broken into smaller layers, fluxes for each layer can be summed
to obtain the overall flux of the colorant.
Figure 7: Kubelka Munk absorption and scattering theory.
16
In this theory, as the light goes farther down the colorant layer, the
magnitude of downward flux (J) is decreased due to absorption and scattering of
the layers. The scattered portion of downward flux (J) will be added to upward
flux I. Similarly a portion of upward flux I is absorbed and added to downward flux
(J). Using differential equations, the downward and upward fluxes can be
represented as:
(2)
di = -(S + K) I dx + Sj dx
dj = -(S+K) J dx + Si dx
where K represents the absorption coefficient and S represents the
scattering coefficient. If P is the ratio of J to I, the above equation can be re-
written as:
(3)
2
)/()/()/(I
dXdIJdxdJIdx
IJddxdp −
==
Re-arranging the above equation and applying boundary conditions
results in:
(4)
∫ ∫ =+−= 20 )(2 SppSKS
dpdx RRg
x
By solving the above equation for R, the famous Kubelka Munk equation
17
(5)
)coth()]coth([1
,
,
XSbbRaXSbbaR
Rg
g
λλλλλ
λλλλλλ +−
−−=
is obtained, where a equals 1+K/S and b equals (a2 -1)1/2 . To solve for the
two unknowns (K and S) in the above equations, two equations are solved by
measuring two ink reflectance samples.
Measurement Requirements: the Kubelka Munk model only requires
measurement of individual inks and the blank medium. If there are N inks in the
system, this model requires only N+1 measurements. The model assumes that
ink and media interactions are homogenous. This model is widely used in the
paint industry.
The accuracy of the model is fairly good in predicting hue; however, it has
problems in predicting chroma of two or more inks overlapped [15].
Neugebauer Model ([18], [20], [21], [15])
The monochrome Murray–Davies model was extended to work for colour
cases and to handle multiple inks by the 1937 landmark work of H. E. J.
Neugebauer ([18]). The Neugebauer model predicts the reflectance of multiple
colorants by summing the products of the fractional area coverage of each
colorant and its reflectance at full area coverage:
(6)
∑=
=N
iii RaR
1,λλ
18
N represents the n Neugebauer primaries at maximum ink coverage. For
instance, for a 3-colorant system, CMY, there are 8 primaries: medium, single
separations (C, M, Y), two-colour overlap (CM, MY, CY), and three-colour
overlap (CMY=K). ai is the area coverage of each primary. There are two
common assumptions used when calculating the area coverage. The Demichel
model, [19], assumes that the half tone dots are printed randomly on the
medium. For instance, the area coverage of two inks in a 3-ink system is no more
than the joint probability of these two inks. The equation below shows the area-
coverage calculation using the Demichel model. C represents the percentage of
cyan ink from maximum coverage. Similarly, M and Y are defined as the
maximum area coverage of the magenta and yellow.
(7)
CMYaYMCa
YMCaYCMa
YMCaYMCaYMCa
YMCa
cmy
my
cy
cm
y
m
c
w
=
−−=
−=−=
−−=−−=−−=
−−−=
)1)(1(
)1()1(
)1)(1()1()1()1)(1(
)1)(1)(1(
The second dot area coverage, Dot-on-Dot, assumes perfect dot
placement overlap.
Similar to the Murray-Davies model, the linear interaction assumption fails
for the Neugebauer model because of optical and physical dot gain. In 1951,
19
Yule and Nielsen introduced a method to model nonlinear interaction of ink and
medium [20]. They showed that the nonlinear relationship between measured
and predicted reflectance could be well described with a power function. Based
on their result, they introduced a modification to the Murray-Davies model as:
(8) npaperi
ni
n RaRaR /1,
/1%100,
/1 )1( λλλ −+=
where n is a parameter to describe the behaviour of light spreading in the
medium. Typically, a nonlinear optimization is used to find the best Yule-Nielson
value. By applying Yule-Nielson theory to the Neugebauer model, the Yule-
Nielson Neugebauer Model is obtained:
(9)
∑=
=N
ii
ni
n RaR1
,/1/1λλ
To improve on the accuracy of the Yule-Nielson Neugebauer model, the
ink space can be measured at a higher resolution (larger number of cells). This
extension is referred to as the Yule-Nielson Cellular Neugebauer (YNCN) Model
[24].
Measurement Requirement: If there are K colorants (inks) in the system
there are 2K Neugebauer primaries and, thus, 2K measurements are needed. By
adding a larger number samples to the system (r samples), there will be rK
20
measurements required. This model assumes a linear relationship between
reflectance and coverage percentage in a space similar to log space.
Challenges of Ink Modelling and Printer Characterization
The Yule Nielson Cellular Neugebauer (YNCN) model is the most
accurate model for characterizing the printer gamut in spectral space. However,
as was discussed earlier, if there are K inks in the system with r samples along
each ink dimension, there will be rK samples to be measured. This means the
number of measurements required for this model grows exponentially as we add
more inks or try to increase the number of samples.
For spectral reproduction purposes, there is a need for a large number of
inks in the system (around 9 or 12) and Tzeng et al. [26] have only extended the
YNCN model to a 6-ink system accurately. One challenge would be to be able to
extend this model to an N-ink printer system where N can be as large as 9 or 12
dimensions.
To reduce the complexity of the model, three concepts are considered in
this paper. First, linearization of each ink before printing the training patches can
be used to keep the Neugebauer cells in uniformly spaced locations and further
reduce the necessary number of steps per colorant. Second, the physical
constraints of the paper, such as the amount of ink it can reliably absorb, can be
used to reduce the potential patches to only the patches that are physically
possible to be printed. The third approach is based on a smarter transformation
21
of the printer gamut before interpolations are applied. This approach is a
sophisticated version of the linearization method discussed in the first approach.
Smarter Sampling: Uniform Sampling in Perceptual Colour Space
In essence, the Cellular Neugebauer model is a piecewise linear model,
and the Yule-Nielsen correction reduces nonlinearity related to dot gain, but does
not capture all of the possible curvature caused by ink interactions, etc. One
method to capture all the curvatures is to increase the sampling range. Using
Taylor series expansion, we can expect that, if we increase the sampling size of
the gamut indefinitely (distance between neighbours 0), the correlation
between two very close neighbours (ink combinations) can be represented
linearly.
As discussed earlier, increasing sampling size is not practical, so another
method can be to find a better process to capture the non-linearity between input
ink combinations and output spectral reflectances.
Linearization is a common transformation method used to improve
accuracy of characterization methods for the output of electronic displays, such
as CRT and LCD devices [1]. The intent of this method is to remove some of the
non-linearity between input channels and output performance by linearizing input
channels against the output performance of the desired device.
The original design of YNCN model calculates the weights used for the
interpolation based on variation in the input (ink density) and not what the
variation in the input channels (ink densities) can cause in the output spectral
22
reflectance. By linearizing the input channels against the output reflectances, the
weights for YNCN interpolation can be better adjusted so that fewer points are
required to be used in interpolation. Raja Balasubramanian [87] introduced a
method of linearizing the YNCN sampling to improve the modeling performance
and reduce number of prints required.
Physical Constraint: Ink Limiting
Printing substrates commonly have a certain ink limit beyond which the
page is too saturated to print. In the inkjet realm this leads to issues such as
cockle, bleed, dry time and gloss non-uniformity. It is not reasonable to print and
measure patches that violate the ink limit of the substrate medium. This
observation can be used to significantly reduce the number of patches required
to measure for the YNCN model. By imposing the constraint that it is not
necessary to print or measure patches that violate the ink limit, the number of
data points to measure for the model can be reduced by up to 97%. The
following is a mathematical analysis of the effect of ink limiting on the number of
training data points that can be printed without exceeding the physical ink
limitation of the paper.
In an inkjet printer, the maximum dispensable weight-per-unit-area Wi for
each colorant i is defined by factors such as drop size, nozzles per inch, and
number of passes. This value varies for each ink, and is generally between 50%
and 100% of the overall ink limit of the medium. The "percent under limit" for
each ink i is defined as Ui = Wi / InkLimit.
23
Given the number of colorants, k, the number of steps per colorant, n, and
the percent under limit, Ui, the complexity subject to the ink-limit constraint can be
computed. For simplicity, it is assumed that U = min(Ui) for all inks, which will
error on the side of over-estimating the complexity.
In the case where U = 1, the printer is capable of delivering exactly the
media ink limit with each ink individually. In two dimensions, the valid sample
space is a triangle defined by (0,0), (n,0), (0,n) as shown in Figure 8.
In general, the space of printable patches can be represented by a k-
simplex (hyper-simplex) defined by the origin and the points along each colorant
axis at a distance of n. The area of such a region is [80] :
Figure 8: Valid Patches for 2 inks with U=1.
(10)
!1 knA
k
U ==
In the case where Ui is between 0.5 and 1, the valid space is a hyper-cube
with sides of length n, and one corner removed by the ink-limit hyper-plane, as
shown in Figure 9.
24
The volume of this region can be computed by calculating the area of the
k-simplex formed by the ink-limit hyper-plane and subtracting the corners that are
outside the printable hypercube, resulting in the following equation:
(11)
( ) ( )kk
U nUnk
kU
nA −−== !)1,5.0(
Figure 9: Valid Patch Space for ½ < U < 1.
For values of U less than 1/k, the ink-limited hyper-plane does not
intersect the dispensable ink hyper-cube, so the complexity reverts to nk.
Computations for 1/k < U > 1/2 are more complex, and are not generally needed
since the colorants are generally defined with U ≥ 1/2.
To determine how effective this constraint is on reducing the number of
measurement points required for the YNCN model, this constraint was
implemented on two training sets, one on glossy photo media and the other on
plain media. In the case of glossy media, 4 steps along the primaries on an 8-ink
25
printer required 6048 patches – a reduction of 91% over nk. On plain media, the
same test required 2175 patches – a reduction of 97%.
The cause for the difference in savings between glossy and plain media
was related to a difference in the definition of the primary ink axes on those two
media. One of the inks was not intended for use on glossy media, so its
linearization table on glossy media was defined such that very little ink would be
dispensed even at 100% fill. As a result, the U-value for that ink was very low
and no clipping occurred in that dimension.
By applying these constraints, the number of training data points is
reduced substantially, which can cause some points in the printer gamut not to
have all the neighbours required for interpolation. The next section discusses
how the YNCN model can be modified to handle cases that do not have all the
points for the interpolation.
Handling the Missing Points (Neighbours)
The YNCN model is based on interpolation between neighbouring
primaries, some of which may have missing data because of the ink-limit
constraint. This poses a challenge for the interpolation operation.
To enable the YNCN model for N inks with missing neighbours, the
weights for all neighbours are computed, the sum of weights for the missing
neighbours is recorded as a "missing score" or M-score value for each
interpolated point. The missing weights are then set to zero and the remaining
26
weights redistributed using a factor of 1/sum(weights) to re-normalize the sum
back to 1.
To redistribute the weights more accurately, the original Neugebauer
model is modified so that weights are calculated based on the linearized distance
of the neighbours in CIELAB colour space. For instance, if the variation along the
yellow primary is smaller than the gray primary, neighbours of the gray primary
will get a larger portion of the weights from missing neighbours.
The second method that was considered handling the missing points was
to extrapolate Neugebauer primaries so that the training data set is populated
enough to cover missing neighbours for a given data set. The idea is that if the
printer spectral gamut is extrapolated slightly, some of the missing points might
be recovered and redistribution of interpolation weights avoided.
The complexity of typical extrapolation algorithms grows rapidly as the
dimensionality of the input data grows. A linear extrapolation method, applied to
one ink at a time, was used in this investigation. For a missing neighbour, P, in
an N ink printer system, N separate interpolations are calculated considering one
of the N inks at a time, resulting in N separate spectral predictions. For each
wavelength, weighted sums of the N predicted reflectances are used to calculate
the predicted reflectance at the same wavelength for the missing neighbour.
27
Smarter Transformation and More Advanced Interpolation Method
One of the other constraints of using the YNCN model, besides requiring a
large number of training points, is the requirement for uniform sampling along
each axis.
Also, as it was discussed earlier, the Yule-Nielson factor was introduced to
remove some of the non-linear interaction of ink and medium. Other techniques
such as ISOMAP can be considered to better capture any nonlinearity in the ink
and medium and thus improve the accuracy of the interpolation.
Improving Printer Characterization using TPS interpolation based on Manifold Transformation
One method considered in this study was to use a printer model based on
Thin Plate Spline (TPS) interpolation. This model has the advantage that the
number of training points and the computational requirements grow much more
slowly than in the case of the YNCN model. In addition, TPS does not require
training data to be sampled on an evenly spaced grid.
TPS can be used to find a continuous function that maps between the set
of inks and each of the output dimensions. For instance if the output spectral
reflectance of an 8-ink printer is measured from 380nm to 730nm with a 10 nm
sampling, TPS is used to create 36 separate functions mapping from N input
dimensions (if there are N inks in the system) to each reflectance wavelength in
10mm increments from 380nm to 730nm,
28
Many spaces appear to have a high dimensionality in a linear space, but
actually have lower intrinsic dimensionality as the Swiss Roll example discussed
earlier. It is possible that the lower dimensional space has a more linear
correlation with the input values that created the space.
One approach that is examined was to improve the printer modeling
algorithms by transforming the output space (printer gamut) to a space that has a
simpler correlation to the input data (ink densities). After the transformation is
applied, TPS interpolation is used to interpolate between input ink densities and
transformed printer gamut data points.
The new interpolation method has the following steps:
1. Find the geodesic distances of spectral reflectances in a printer gamut
2. Map the printer gamut into a new space (typically of lower dimension)
using the geodesic distances (ISOMAP Technique)
3. Create continuous functions between input ink combinations and
transformed gamut space
Data Collection
An 8-ink printer with the following inks was used to study performance of
the model: cyan, magenta, yellow, light cyan, light magenta, black, gray, and light
gray. The results are based on 6048 patches for training and 939 patches for
testing. The patches were printed on glossy media with an ink limit of either 1.5
or 2 drops of ink (depending on the ink type).
29
A GretagMacbeth Spectralino [5] was used to measure the spectrum
reflectance of the printed patches consisting of 10nm sampling from 380nm to
730nm. The Spectralino has an accuracy of around 0.30 ∆E94 between two
different sets of measurements under the D50 illuminant. The printer has an
average 0.75 ∆E94 page-to-page variation (including instrument variation).
Implementation
Drops of inks which are intended to fall on top of each other during the
printing process can fall on top of each other (Dot-on-Dot), beside each other
(dot-by-dot) or have a more random placement (Demichel). There are modeling
methods to capture each scenario.
In this section, both Demichel [23] and Dot-on-Dot [81] models were
implemented. The Dot-on-Dot model assumes perfect dot placement during
printing, whereas the Demichel equations assume a more random dot
placement, which is more suited to ink-jet printers with a half toning process done
to redistribute the dot placement.
Three different methods for handling missing points in the model (missing
neighbours) were implemented and studied. The three models are as follows:
Distributing weights for missing neighbours with and without some linearization or
extrapolating Neugebauer primaries to fill in missing neighbours as much as
possible. These three approaches are referred to as Lin=0, Lin=1, Extrap=1.
In order to find the proper Yule-Nielson correction factor, a search was
done to find the best value. The search used starts with incrementing the Yule-
30
Nielson factor and stops if the prediction accuracy is within 0.10 ∆E94 for the
subsequent Yule-Nielsen values.
Results
Results: Modified YNCN (Smarter Sampling and Dealing with Missing Neighbours)
Table 1 shows the accuracy of the model for both Dot-on-Dot and
Demichel dot placement approaches. The results are shown both in ∆E94 for D50
illumination and spectral difference calculated as Root Mean Squared (RMS)
difference between prediction and input reflectances.
Table 1 shows that the Demichel model is more accurate than the Dot-on-
Dot model for predicting reflectance of multi-ink systems and modeling. The data
also shows that YNCN performance is improved by linearizing the training data
before calculating the interpolation points. Figure 10 shows that the main error in
the YNCN prediction comes from the data points that have too many missing
neighbours. Table 2 also shows that correlation of the error with the missing
neighbours when different methods are used to redistribute weights of the
missing neighbours. The important conclusion taken from Table 1 is that a simple
linear extrapolation of the training data point to recover as many as missing
neighbours is more effective than redistributing the YNCN weights for the missing
neighbours.
31
Table 1: YNCN Performance for an 8-ink printer with missing Neighbours. Mean, Max and Std represent average, maximum and standard deviation of the error respectively [68]
∆E94 RMS
Demichel, YN=5.2, Lin=0 Mean 2.43 0.0072
Max 10.92 0.048
Std 1.49 0.0057
Demichel, YN=5.1, Lin=1 Mean 2.31 0.0064
Max 10.73 0.0613
Std 1.34 0.00542
Demichel, YN=4.1,Extrap=1 Mean 1.48 0.0047
Max 5.12 0.0284
Std 0.81 0.00299
Dot-on-Dot, YN=7.5, Lin=0 Mean 8.54 0.0363
Max 37.46 0.2848
Std 5.42 0.0367
Dot-on-Dot, YN=7.3, Lin=1 Mean 7.87 0.0313
Max 29.38 0.257
Std 5.08 0.0326
Dot-on-Dot, YN=7.1,Extrap=1 Mean 3.00 0.00584
Max 11.75 0.0471
Std 1.56 0.00465
32
Figure 10: Correlation of missing neighbours and error for Extrap=1 method, R2=.38.
Table 2: Correlation between missing score (percentage of missing neighbours) and error in the model (∆E94)
Method Correlation
No Extrapolation, No Linearization (Lin=0) 0.63
No Extrapolation, with Linearization (Lin=1) 0.48
Extrapolation, No Linearization (Extrap=1) 0.38
33
Result: Manifold Based Printer Model
All the existing printer modeling algorithms including the YNCN model are
based on different interpolation techniques. The accuracy of most interpolation
techniques is improved if the input data has a simple correlation with output data.
For this reason, the Yule-Nielson factor is used in the YNCN model to transform
the spectral output to a space that has a more linear relation to the input ink
densities.
Figure 11 shows a dimensionality analysis of the printer spectral gamut of
the 8-ink printer system before and after applying a Yule-Nielson value of 2
(Note: Using a Yule-Nielson factor of 2 is effectively applying logarithmic
transformation of base 2). Principal Component Analysis (PCA) is applied to the
data set, and residual variance between reconstructed data from PCA basis and
original data is plotted in the Y axis. The X axis represents the number of bases
used to reconstruct the input data.
The figure shows that, after applying a non-linear transformation to the
printer gamut, the dimensionality of the gamut is reduced to almost 4 rather than
6 dimensions.
34
Figure 11: Residual error of spectral recovery as a function of reflectance dimensions. The data is for spectral measurement and spectral measurement after a logarithmic transformation.
Knowing that the Yule-Nielson transformation was effective in reducing the
dimensionality of the printer gamut, the ISOMAP transformation was used to test
whether this transformation can be optimized and, thus, improve the printer
modeling algorithms. Figure 12 shows that after the ISOMAP transformation is
applied to the printer spectral gamut, the dimensionality of the system is reduced
to 3 dimensions. The figure also shows that applying a Yule-Nielson
transformation after ISOMAP transformation does not help with the
dimensionality reduction.
35
Figure 12: Residual error for spectral recovery of the 8-ink printer spectral gamut after ISOMAP transformation. ISOMAP (LOG) shows when a Yule-Nielson of value 2 is applied to the ISOMAP transformed gamut data.
The first important observation that can be drawn from this comparison is
that the printer spectral gamut may not be as complex as it has been observed
by looking at the gamut space directly. Knowing how neighbouring points that
created the printer spectral gamut are connected to each other can be used
towards space complexity reduction of the printer gamut. The second
observation is that knowing which ones are the true neighbours of a point on the
printer gamut and how they are connected to the point of interest, more accurate
weights than those yielded by the YNCN model should be calculated for the
printer model.
To evaluate the accuracy of the new modeling algorithm, TPS
interpolation was used to interpolate between ink density and output spectral
reflectance before and after the ISOMAP transformation. Since the ISOMAP
calculation has higher time complexity than the YNCN algorithm, the training
36
sample size was reduced to half of the training sample size used for the YNCN
model in order have a similar or slightly faster modeling algorithm. The samples
were selected randomly from the ones used in the YNCN model.
Figure 13 and Table 3 present the performance of the TPS interpolation to
predict the spectral gamut directly and after ISOMAP transformation. The results
are compared against the YNCN model. The data shows that applying TPS
interpolation to predict the spectral printer gamut directly, given ink densities, has
poor performance. The performance is slightly improved if the interpolation is
applied after the printer gamut data has gone through a logarithmic
transformation. This can be explained by knowing that the transformed space
has a lower dimensionality (as presented in Figure 11).
On the other hand, applying TPS interpolation after the ISOMAP
transformation is as accurate as the YNCN model, when only half of the training
data points were used for this new model.
Table 3: Performance of Geodesic (ISOMAP) and Linear based modelling of the 8-ink printer system using TPS (Thin Plate Spline) interpolation. ISOMAP (LOG) and Spectral (LOG) represents TPS interpolation applied to each space after they have gone through a logarithmic transformation (which is similar to having the Yule-Nielson factor equal to 2).
RMS ∆E94
Mean Max Mean Max
YNCN 0.0047 0.0284 1.48 5.12
Spectral 0.0097 0.1027 3.1 47
Spectral (LOG) 0.0086 0.09 2.1 23
ISOMAP 0.005 0.0629 1.99 6.71
ISOMAP (LOG) 0.0049 0.068 2.1 7.2
37
Figure 13: Performance of Geodesic (ISOMAP) and Linear based modelling compared against the YNCN model. The vertical axis represents the average error calculated as the Root Mean Square difference between predicted reflectance and measured values.
Printer Characterization Performance
0
0.002
0.004
0.006
0.008
0.01
0.012
YNCN Spectral Spectral (LOG) Isomap Isomap (LOG)
Mean
RMS
38
CHAPTER 4: NUMBER OF PRIMARIES
One of the main goals in spectral reproduction is to reduce metamerism
by matching each input spectrum as closely as possible, while requiring the
minimum number of primaries. In the printing industry, to make a good spectral
match, the printer gamut is expanded by adjusting the chemistry of the inks, and
especially by increasing the number of the inks used in the printer [28], [29], [30].
Similarly, in cameras and display technologies, it is now common to make the
system with more than the traditional three primaries (sensors, LEDs, filters or
phosphors) to further reduce the degree of metamerism [32], [33], [34]. The focus
of this section is to determine a lower bound on the number of primaries needed
to do a reasonable job in spectral printing.
Calculating Data Set Complexity
A simple approach to measuring the complexity of a data set is to assume
that the number of channels needed in a system is limited and relates to the
underlying dimensionality of the captured data in a linear space. Approaches
such as Principal Component Analysis (PCA) and Independent Component
Analysis (ICA) are widely used in research-related spectral data dimensionality
where it is agreed that a small number of basis functions is adequate to
represent a high dimensional data set accurately.
39
For instance, in studies of Munsell Colours, Eem et al. [35] proposed four,
Maloney [36] proposed five to seven, Burns [32] suggested five to six, Parkinnen
et al. [37] and Wang et al. [38] proposed eight, and Hadeberg et al. [39]
recommended as many as 18 basis functions to represent the data accurately.
Even though there are variations in their findings -- because of having different
thresholds for measuring the similarity between the original and matched spectral
data -- all the authors used similar techniques to analyze the complexity of a data
set.
Recently the focus of the research has been on proposing the number of
needed basis functions plus their reflectance characteristics [27][30]. The
advantage of these approaches is that the proposed basis functions have similar
reflectance or sensitivity characteristics to physically available solutions. Using
these methods, researchers can optimize the characteristics of the primaries in a
system to better capture or reproduce a reflectance with a minimum number of
primaries.
Rotated PCA Basis
PCA provides a method to determine the dimensionality of the spectral
sample population [40]. PCA has been widely used in colour-related applications
[41], [42]. The main assumption of this technique is that the set of sampled
vectors (A) is multivariate normally distributed in the original dimensionality
(typically 31 dimensions for spectral analysis). The linear combinations of the first
p eigenvectors should describe the entire set of Aλ if the original was created by
p linear basis, i.e.:
40
(12)
EB== ∑=
p
iiisample ebA
1,, λλ
Where eλ,I is the ith eigenvector and bi is the corresponding coefficient to
reconstruct a sample.
One of the drawbacks of using techniques such as PCA is that the
returned basis functions do not necessarily correlate to the physical
dimensionality of the data. For instance Di-Yuan et al. [27] observed that the
PCA basis functions representing a set of painting reflectances have negative
values which do not correspond to the actual ink amounts (Figure 14).
Figure 14: The six eigenvectors obtained from the still life painting by Di-Yuan [27].
One approach to overcome the negative values of the basis function is to
develop a linear transformation of the basis functions so that the new functions
41
have all positive values. Ohta [43] proposed running a regression search method
to find the best transformation of the PCA basis functions that reconstructs the
spectral data set accurately and which has two main properties: all the basis
functions have positive values and the concentration matrix should have all non-
negative entries. After the transformation is applied to the PCA basis, the bases
are not necessarily orthogonal to one other.
PCA Eigenvector without Sample Mean
Di-Yuan et al. [27] proposed eigenvector reconstruction of PCA without
removing the sample mean to find basis functions that are closer to realistic
primary characteristics. Using PCA basis functions a spectrum can be
reconstructed as:
(13)
∑=
+=p
isamplemeaniisample AebA
1,,, λλλ
The sample mean used in PCA is only a statistical parameter which
specifies the average data set behaviour. The sample mean does not represent
any physical colorant. Also, since the eigenvectors are the only clue leading to a
set of possible colorants, the sample mean must be excluded to maintain the
transformation relationship between eigenvectors and the set of possible
colorants which is specified by the above equation.
42
Di-Yuan et al. ran an experiment on a data base of paintings created by
six independent acrylic-paints. 126 samples were measured. Figure 15 shows
the properties of the 6 paints.
Figure 15: The six acrylic-paints used for generating the sample population. The vertical axis shows the K/S factor of Kubelka Munk theory, a representation of reflectance (absorption/scattering).
The initial six rotated eigenvectors without removing the mean are shown in Figure 15.
Table 4 shows the accuracy of using the first 6 basis functions of PCA in
reconstructing spectral reflectance. The data shows that the 6-basis-driven
analysis using the described method can reproduce the intended set of data
quite accurately.
43
Table 4: Colorimetric accuracy of the six estimated colorants for the painting database
DeltaE94 Mean 0.22 STD Deviation 0.16 Maximum 0.92 Minimum 0.02
As shown in Figure 15, the estimated colorants are not similar to the
measured reflectance of the 6 paints. There is also a colorant spectrum (thick
dotted line) with various absorption bands across the visible spectrum. None of
the predicted colorants represents a flat (neutral) spectrum either. The neutral
colorant with an approximately flat spectrum can be approximated using a linear
combination of the other five estimated colorants. To include the neutral colorant
in the predicted basis, a new constraint was proposed to first estimate the neutral
colorant using linear regression to fit the perfectly flat spectrum by the six
eigenvectors. Then, the most significant fix eigenvectors were rotated individually
to a non-negative representation. The curves of the new 6 reflectances are
shown
44
Figure 16: The estimated colorants (solid lines) and the original colorants (marked as start) used in the painting data base.
Multi-Peak Primaries [30]
As discussed earlier, in order to match the characteristics of predicted
primaries to those of real primaries, a constrained search was used to guarantee
positive reflectances or sensitivities [27][43].
Previous approaches worked with reasonable accuracy in lower
dimensions. However, the modulation of the characteristics of the primaries is
proportional to their order, i.e., additional vectors have an increasing number of
peaks. It is preferred that, as the number of primaries increases, the algorithm
can find the primaries that have non-negative values so that their individual
sensitivities can be concentrated in distinct regions of the visible spectrum [30].
45
For example, in a trichromatic camera, the sensors are commonly chosen to be
red, green and blue so the peaks are almost evenly spaced in the visible
wavelength. Finally, it is preferred that the transformed vectors should ideally
span the same space as the original.
Hardeberg et al. [30] proposed a search algorithm based on varimax
rotation preferences [31] to develop a transformation that meets the above
preferences. This transformation can be represented as a matrix transformation
B, such that matrix B takes an initial basis to a non-negative basis. The algorithm
searches for an orthogonal rotation to B that maximizes the preferences
(varimax) criterion. The preference (varimax) metric is a combination of distance
of the peaks plus width of the peaks. For instance, it can include the preference
that the peaks of sensors in a camera system be spaced equally and have
similar widths. The preference metric can also be more specialized. For instance,
it can be adjusted to allow wider peaks at lower wavelengths.
Hardeberg also emphasizes that most of the synthetic analysis that is
done on the number of primaries and their characteristics does not consider the
noise that exists in the real system. Adding more primaries improves
metamerism if the system is noise free. However, if each primary has noise in
reproducing or capturing a reflectance, more primaries means a higher noise
level and thus lower reproduction accuracy. He created the data base by adding
two types of noise to the modelled reflectances:
(14)
46
quantshot
p
iiisample nnebA ++= ∑
=1,, λλ
Nshot represents the noise in generation and reflection of light, and Nquant is
the noise associated with quantising the simulated responses. Figure 17
compares the effect of noise in a camera system plus the effect of imposing
preferences on sensor characteristics. Piche and varimax are two ways of
imposing the preference that Hardeberg considered on a data base of 1269
Munsell reflectances.
Figure 17: The effect of increasing the number of sensors with 12-bit quantization and 1% shot noise.
Compared to synthetic noise-free systems, Hardeberg found that when
the real physical constraints of a system are considered, many fewer sensors are
required to get the maximum reconstruction from the system.
47
Analysis of Primary Characteristics
In this section, a method is proposed to evaluate the extent to which
spectral reproduction accuracy is improved as more and more primaries are
used. The effect of primary reflectance characteristics on device output gamut
performance in spectral space is also considered.
Primary Selection
One purpose of this experiment was to understand the effect of primary
reflectance characteristics on the accuracy of spectral reproduction. This data
can help better define what changes can be done with respect to a given set of
primaries in order to optimize the accuracy of spectral reproduction and reduce
metamerism in a system. Reflectance characteristics that will be discussed are
the effects of variations in the number of primaries, percentage of overlap
between each primary and degree of smoothness of each primary reflectance.
To compare the performance of the available primaries in the industry
against what can be used as an optimal set, two general sets of primary
characteristics were used in this study. One set was based on reflectances of
real inks, and the other set was synthetic primaries (ink or filter light reflectance
depending on the device model used). Both synthetic and actual measurement
data were used to make the result less dependent on a specific ink selection.
The real ink reflectance measurements were based on actual prints of pigmented
inks. The following 9 inks were used: orange (O), cyan (c), magenta (m), yellow
(y), Green (Gr), violet (V) and black (K), light magenta (LM) and light cyan (LC).
48
Three types of synthetic primaries were also used. The first type of
reflectance used was based on square-wave reflectance as shown in Figure 18
where the edges are sharp, and thus resembles sub-sampling of the spectral
reflectance. The 3 inks, as shown in Figure 18, cover the visible wavelengths 380
to 730, and are non-overlapping. The 6 inks were created by subdividing each
ink in the 3-ink model into two separate square waves. The set of 9 and 12 non-
overlapping inks were created similarly. The white of the print medium was taken
to be the ideal white with 100% reflectance at all wavelengths.
Figure 18: 3 Square wave synthetic ink reflectances covering 380 to 730 nm equally.
The second set of primaries studied has a more gradual transition from a
non-absorbance region to the area of reflectance absorbance. This set of
reflectances was used to compare the effect of having a tail (gradual transition
between the absorptive and non-absorptive regions of each primary) on the
accuracy of spectral reproduction.
49
Two sets of primaries with tailed overlap were considered as shown in
Figure 19 and Figure 20. One set was based on modified square-wave
reflectances with longer tails with the other set more sinusoidal (Figure 20).
Figure 19: Two primaries with non-smooth tailed endings. The reflectances of these primaries have a gradual transition between absorptive and non-absorptive regions.
Figure 20: Two primaries with smooth tailed endings. The reflectances of these primaries have a smooth gradual transition between absorptive and non-absorptive regions.
50
Figure 21: Reflectance of real inks measured. The curves show the smooth and long tails that are common for real reflectances.
Another contributing spectral reflectance characteristic of primaries
studied was the percentage of spectral reflectance overlap. To evaluate the
possible benefits of overlap for each type of reflectance, a set of 4 different
reflectances in each reflectance type with 0%, 10%, 20% and 40% overlap was
used. Figure 22 and Figure 23 compare two different sets of square-wave
functions with different degrees of overlap.
Figure 22: Square-wave reflectance functions with 0% overlap.
51
Figure 23: Square-wave reflectance functions with 20% overlap.
Device Characteristics
To evaluate the effect of the interactions between the primaries on the
spectral reproduction accuracy, two types of device models were considered.
Two device models, denoted LP (linear projector) and TK (Tzeng simple-Kubelka
Munk), were used to predict the spectral reflectance resulting from printing or
displaying a given primary combination. For LP, the displayed reflectance was
assumed to be a linear combination of the primary reflectances. The equation
below expresses how the model works:
(15)
Rλ = [ΣciRλ,i] = C(1xn)Rλ(nx1)
R λ,i is the reflectance of primary i at 100% density, and ci is the area
coverage. Rλ(nx1) represents a matrix of size nx1 of reflectances at wavelength λ.
The LP model assumes that the primaries mix linearly in a subtractive
colour mixing system and there is no non-linear interaction between the primaries
[64]. The advantage of LP is that the ink separation algorithm to reproduce an
input reflectance becomes a straightforward linear algebraic operation. In
52
addition, this model is quite similar to how output for displays (e.g., monitors and
projectors) is modelled [1] and, because of that, the data from the LP model can
also be used to evaluate the performance of displays using different primary
characteristics.
The printer model, TK, introduced by Tzeng et al. [27], [29], was used to
mimic the real ink and media reflectance. The following equations were used to
predict the reflectance of multi-ink printing system:
(16)
Rλ = (Rλ1/w
paper - ψλ,mixture) w
ψλ, mixture = Σci Riλ
ψλ = Rλ1/w
paper - Rλ,i1/w
where w is the non-linearity weight similar to Yule-Nielsen factor [21] and
Rλ,i is the reflectance of the ith ink as a function of wavelength.
Evaluation Method
To determine how the number of primaries affects the accuracy of spectral
reproduction in terms of reproducing spectra, spectral matches for 3-, 6-, 9- and
12-primary devices (LP and TK models) were calculated. The performance of
these synthetic primaries was compared against real ink spectral reflectances in
order to understand how optimized the reflectances of the existing inks are. For
case of the reflectance of actual inks, the 3 inks considered are the most
common 3 inks used in practice, namely, cyan, magenta, and yellow. For the 6-
53
ink case, the initial 3 inks are retained, and 3 more complementary inks are
added, namely, orange, green, and violet. light cyan, black and light magenta are
added to these 6 for the 9-ink case.
However, the complexity of the conventional methods for evaluating
gamut performance of a printer grow exponentially as the dimensionality of the
input data increases (The analysis is discussed in more detail in Chapter 4). One
method to indirectly evaluate spectral gamut size of a printer system which
includes a set of primaries and a printer model is to evaluate how accurately the
printer system can reconstruct a database of input reflectances. This method
depends on spectral gamut mapping algorithms to estimate the printer gamut
size.
For the LP (Linear Projector) model -- since the primaries interact linearly
with each other -- gamut mapping becomes a simple linear projection operation.
The equation below shows the process of deriving the closest primary
combination to reconstruct an input reflectance:
Rλ = [ΣciRλ,i] = C(1xn)Rλ(nx1) (17)
C(1xn) = (Rλ(nx1))-1 Rλ
However, when the primaries interact non-linearly, gamut mapping
becomes more complicated.
54
Hierarchical Search (HS) Gamut Mapping Algorithm
For the TK printer model -- since the primaries interact with each other
and the media non-linearly -- the gamut mapping process cannot be done using
a linear projector operation. There are some proposed methods to handle gamut
mapping in spectral space that will be discussed in Chapter 4. However, most of
these spectral gamut mapping algorithms try to improve their time or space
complexity at the expense of accuracy.
To understand the effect of each primary characteristic on gamut
performance independent of the gamut mapping algorithm performance, results
were based on a search-based mapping algorithm. This algorithm is based on
hierarchical search in ink space where the search is done in subdivisions of ink
combinations. Let the set of the subdivisions of ink space be M, where there is a
spectral reflectance associated with each ink combination, mi, in M. The
algorithm is as follows:
1. Find the closest mi spectrum to a given input point p in spectral
space.
2. Create a grid of ink subdivisions around mi with smaller ink
variation.
3. Go back to step 1 until the grids are small enough. Then go to the
next step.
4. Return the spectral reflectance of mi as the closest point.
55
The main drawback of the search algorithm is that its accuracy highly
depends on the sampling resolution of the system. Because of the exponential
growth in the sampling as a function of the number of inks in the system, a
modification to the search algorithm is needed for printer systems with 9 or more
inks. The modification involves breaking the spectral wavelength range into
segments and running the search for each segment independently. The premise
is that if the reflectance wavelength of interest is segmented into T subsections,
each section can be analyzed separately. To account for inks that have
absorption sensitivity on more than one segment, the neighbouring segments
have 30% overlap in their wavelength. For example working with spectral
reflectances ranging from 380 to 730 wavelengths with 3 segments, the first
segment covers 380 to 520, the second one covers 470 to 640 and the third one
covers 590 to 730 nm in wavelengths.
For each segment, the algorithm only considers the inks that have
absorption sensitivity in that segment wavelength range. Applying this second
filter reduces the number of inks that need to be considered, thus enabling the
search method to have similar sampling rate as for printer systems with fewer
number of inks.
Optimizing the HS Parameters
For the HS algorithm, there are two sets of parameters to optimize. The
first set of parameters represents the maximum number of iterations allowed
during the search and the number of samplings to have along each dimension.
56
The second set of parameters represents the number of segments that the
input reflectance will be divided into and the percentage of overlap between
neighbouring segments. The higher the number of segments, the fewer the
number of primaries in each segment and, thus, the faster the search algorithm
will be. However, smaller segments means more segments that the search has
to deal with. Also, if the segments are too small (cover small range of
wavelength) and the overlap percentage is kept the same, the accuracy of the
model can be low. As a result, a balance needs to be made between the number
of segments and the overlap percentage.
The output of the HS algorithm was evaluated on an output system with
LP primary interaction (linear interaction). Doing so enabled a comparison of HS
performance against linear projection, as well as optimization of the parameters
to get the lowest error from the HS search algorithm.
In this experiment, the first parameter, which is the number of search
iterations, was kept constant for 3, 6, 9 and 12 ink printer systems. For 3 and 6
ink systems, based on the comparison to the LP output model, it was enough to
use only one segment. For 9 and 12 ink printer systems, 1, 2, 3 and 4 segments
in spectral space with 10, 15 and 20 percent overlap for each were tested and
the best parameter was selected. Table 5 represents the performance of the HS
algorithm against projection method for square type inks. The table shows that
after selecting the right search parameters, the HS search method finds results
that are very similar to the direct linear projection method. The parameters are
then used for HS search algorithm for TK printer model.
57
Table 5: Performance of the HS search for 3, 6, 9 and 12 ink printer systems with the LP Printer model. The performance was compared against the linear projection method that returns the true answer. Square wave inks were used in the experiment.
Number of Inks mean deltaE max deltaE min deltaE
3 0.263 1.09 0.14
6 1.09 2.39 0.04
9 1.23 2.61 1.64
12 1.58 3.39 1.69E-01
Evaluating Metamerism
Root Mean Square (RMS) difference between two reflectances is one of
the common metrics used for evaluating the similarity between two spectral
reflectances. However, RMS (root mean square) does not necessarily represent
the difference that may be perceived by a human observer. As an alternative
measure, the average colour variation calculated as deltaE94 of the two spectral
reflectances found under 11 different lights was used. The 11 illuminants used
were from the Simon Fraser data base [65] shown in Table 6.
Table 6: The 11 different illuminations used in measuring the colour variation of two similar reflectance spectra. This data is from Computational Vision Lab at Simon Fraser University [65].
58
11 illumination types used for delta E comparison Sylvania 50MR16Q (12VDC)---A basic tungsten bulb Sylvania 50MR16Q (12VDC) + Roscolux 3202 Full Blue filter Solux 3500K (12VDC)--Emulation of daylight Solux 3500K (12VDC)+Roscolux 3202---Emulation of daylight Solux 4100K (12VDC)--Emulation of daylight Solux 4100K (12VDC)+Roscolux 3202---Emulation of daylight Solux 4700K (12VDC)--Emulation of daylight Solux 4700K (12VDC)+Roscolux 3202---Emulation of daylight Sylvania Warm White Fluorescent (110VAC) Sylvania Cool White Fluorescent (110VAC) Philips Ultralume Fluorescent (110VAC)
Scene Data Base and K Means
To evaluate the effect of each ink on spectral gamut coverage, the scene
reflectances from the Simon Fraser University (SFU) database were used as
target reflectances to reproduce. There are 1350 individual reflectances in the
database. However, time complexity of the HS ink separation method grows
exponentially as the number of primaries used in the system increases. To speed
up the evaluation process, the number of scene reflectances used in the
experiment was reduced to a smaller set. In order to have the smaller reflectance
database better represented, the original database was classified to subsets. K-
means clustering was used to classify the original database to smaller subset
[83]. This clustering technique includes four steps: (1) Select k initial start points
as cluster centres; (2) Calculate each pixel's distance to the cluster centre; (3)
recalculate each cluster's centre; (4) Repeat until converged to a stable status.
59
It was found that the SFU scene database can be classified to 80 disjoint
classes accurately enough. One reflectance from each cluster is used to
represent the cluster. All the evaluations in this section are based on the 80
selected scene reflectances. Figure 24 and Figure 25 show the complexity of the
80 selected reflectances. The figures show that in a linear system, if the
primaries are selected optimally, at least 4 primaries are needed to have less
than 2 deltaE94 colour reproduction and very similar spectral reflectances. This is
assuming that the primaries can have both positive and negative reflectances
(which is not realizable in real printers or displays).
Figure 24: Mean Root Mean Square residual error as the number of PCA bases increases.
60
Figure 25: Mean DeltaE94 error between reconstructed reflectance and the database reflectance when different numbers of PCA bases used. DetaE94 is calculated as average detalE under 11 different illuminations provided from the SFU database.
Results
Primary Overlap
In this section, the effect of having different percentages of overlap
between the primaries is evaluated. The result is repeated for each type of
primary reflectance. Figure 26 shows that for the square type reflectances, as the
overlap percentage increases, the accuracy of the spectral matching will
decrease. Figure 27 shows similar behaviour if the performance is evaluated as
∆E94 colour difference under 11 different illuminations.
Figure 28 looks at one reflectance matching using 6 square waves with
0% overlap and 20% overlap. The figure shows that for square type waves, for
which both the centre of the signal and the edges of the signal have similar
coverage (i.e., there is no tail for the signal), having overlap on the primary
reflectances causes a significant drop in reproduction accuracy. This is an
indication of a drop in gamut coverage of the device.
61
Figure 26: Spectral Gamut coverage of square wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on non-linear (TK) model.
Figure 27: Spectral Gamut coverage of square wave ink given 3, 6, 9 and 12 inks evaluated as DeltaE94 colour difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The error is shown as average variation under 11 different illuminations. The printer model is based on non-linear (TK) model.
62
Figure 28: Input reflectance and closest match using 6 ink square waves with 0 and 20 % overlap. The printer model is based on non-linear (TK) model.
Figure 29: Spectral Gamut coverage of square wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on linear (LP) model.
Considering trapezoidal signals (Figure 30) which have a tail (slower drop
in their absorption sensitivity compared to square wave), some level of overlap
improves printer spectral gamut performance. This characteristic holds for a sine
wave signal as well, as shown in Figure 32 and Figure 33.
63
Figure 30: Spectral Gamut coverage of trapezoidal wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square (RMS) difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on a non-linear (TK) model.
Figure 31: Input reflectance and closest match using 6 ink square waves with 0 and 20 % overlap. The printer model is based on a non-linear (TK) model.
64
Figure 32: Spectral Gamut coverage of sine wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square (RMS) difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on a non-linear (TK) model.
Figure 33: Spectral Gamut coverage of sine wave ink given 3, 6, 9 and 12 inks evaluated as Room Mean Square (RMS) difference between the database of reflectances and the closest reflectance matches that fall on the gamut. The printer model is based on a linear (LP) model.
Primary Interaction Model
Figure 35 Compares gamut coverage of the two primary interaction
models for square waves. Previous data showed that for the square wave
65
functions, the more the two primaries overlap the lower the gamut coverage
becomes (Figure 26). One hypothesis can be that if the primaries have optimal
spectral overlap percentage, then the two interaction models (linear and non-
linear) have similar gamut coverage. However, once there is more overlap
between the primaries’ absorption sensitivity regions than what is optimal, then a
non-linear system has better performance. A possible explanation for this is that
when the inks have more than the optimal overlap, there are many irregularities
(non-uniformities) between the matched reflectance and the target reflectance.
When a non-linear system is used, some of these irregularities are smoothed out,
which can result in a more accurate reproduction as shown in Figure 34.
Figure 34: Spectral match of a scene reflectance using a 3-ink system with square wave inks of 20% overlap. The reproduction is shown for linear and non-linear systems. For linear system the reproduction error has RMS value of .11 and non-linear system has .065
Similar behaviour is seen for the sine and trapezoid shape signals that
have longer tails. Figure 36 and Figure 37 show that for signals with not enough
66
overlap between the primaries, a linear model performs better. When there is
enough overlap (e.g. 10% for trapezoid signal or 20% for sine wave), then both
linear and non-linear models have overall similar coverage. On the other hand,
when there is more than optimal overlap between signals, primaries with non-
linear interaction result in a better reproduction than if they had linear interaction.
Typical inks used in the printers are a good example of primaries with long
tails. Based on observations using synthetic inks, the expectation is that these
inks (real inks) would have better spectral gamut coverage in a non-linear system
than a linear system. Figure 38 confirms the expectation, where 3, 6 and 9 real
ink reflectances were used.
Figure 35: Performance of square wave primary evaluated as RMS of match between scene reflectance and the closest match on the system gamut.
67
Figure 36: Performance of square wave primary evaluated as RMS of match between scene reflectance and the closest match on the system gamut.
Figure 37: Performance of square wave primary evaluated as average RMS of match between scene reflectance and the closest match on the system gamut.
68
Figure 38: Performance of real ink reflectances as average RMS of match between scene reflectance and the closest match on the system gamut (both linear and non-linear models are evaluated).
Number of Primaries
The focus of this study was to understand whether the spectral gamut
coverage of the device is improved when more primaries are made available in
an output device.. The result in this section tries to explain how many primaries
are needed in a device to get an accurate spectral colour reproduction system
when the reflectance or absorption characteristics of the primaries are, or are
not, optimized for reproduction purposes.
Figure 39 and Figure 40 show, for close to optimum square wave
primaries, the spectral coverage of a device gamut improves noticeably as the
number of primaries increases (whether the error is measured in RMS or
DeltaE). On the other hand, if the overlap percentage for the same type of
primaries is larger than what is desired (having non-optimum primaries), then the
gamut coverage of the output device does not improve continuously as the
69
number of primaries increase (Figure 41 and Figure 42). The data shows that for
non-optimized primaries, the gain from having a higher number of primaries
(especially after 6 primaries) is cancelled by the noise in spectral reproduction
from having a higher than optimized overlap amount. Similar behaviour is seen
for output devices with linear primary interaction (Figure 43).
Figure 39: Spectral Gamut coverage of sine wave ink at 20% overlap (the better overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis.
70
Figure 40: Spectral Gamut coverage of square wave ink at 0% overlap (the better overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis.
Figure 41: Spectral Gamut coverage of sine wave ink at 40% overlap (the undesired overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis.
71
Figure 42: Spectral Gamut coverage of square wave ink at 20% overlap (the undesired overlap amount) for 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis.
Figure 43: Spectral Gamut coverage of square wave ink at 40% overlap for a linear system. 3, 6, 9 and 12 inks evaluated based on mean RMS difference between the closest match on gamut and the goal reflectance. Mean RMS is on Y axis.
72
CHAPTER 5: SPECTRAL GAMUT MAPPING AND SPECTRAL INK SEPARATION
As the number of independent primaries increases in a system, the gamut
coverage of the system grows. On the other hand when working with data in
higher dimensions, the likelihood of some input data falling outside of the gamut
increases exponentially. A similar situation applies in spectral printing, meaning
that given a medium (paper) maximum reflectance constraint, non-linear
interaction of the inks and wide spectral sensitivity of the inks, it is almost
guaranteed that a given input spectral reflectance falls outside of the printer
gamut. Because of such a high probability of an input point falling outside of the
device gamut, gamut mapping becomes the base of spectral reproduction,
especially for non-linear devices such as printers. This is quite different from the
traditional printing approach where a considerable portion of the data falls inside
the gamut. Because of this, the focus of this portion of the research is on spectral
reproduction, which includes gamut mapping and primary separation (ink
separation), for the printers where the non-linear interaction of inks and media
increases the complexity of the reproduction algorithms substantially.
Another challenge with working in higher dimensions is the complexity of
calculating the gamut. The traditional gamut mapping algorithms work in CIELAB
or other three dimensional spaces. Most of these algorithms calculate the gamut
boundary using methods such as convex hull [28], [45], [46], [47]. The complexity
73
of these methods grows exponentially with the number of dimensions, which
poses another challenge to spectral reproduction.
In this section, a review of different approaches for finding the best ink
combination and mapping a given reflectance on the printer gamut boundary is
presented.
Spectral Reproduction Based on Interim Colour Space
ICC (International Colour Consortium) Profiles describe the colour
behaviour of a particular device by defining a mapping between the source colour
space and a profile connection space (PCS). Typically, this PCS is based on
either CIELAB or CIEXYZ colour space. There are two common types of
mappings. One approach uses look-up tables where interpolation is applied for
the data points between the cells. The other mapping is based on a series of
parameters for transformations, e.g. a 3x3 matrix transformation.
Several approaches have been proposed to extend the idea of ICC to
spectral space. Mitchell et al. [48], [49] proposed a combination of projection plus
a 6-dimensional Lookup table (LUT) PCS for a 6 ink printer. An iterative
interpolation was used to invert the LUT. Their study [48] showed that using an
ink-like basis improves the efficiency of the LUT compared to using orthogonal
eigenvectors generated from PCA.
Berns et al. proposed a new method based on using a coarse LUT to
reduce metamerism [52]. Using this method, in each cell of the LUT there are
multiple ink-combination candidates that have the same colour value (CIELAB)
74
under one illumination but a different colour value under the second source of
illumination. Knowing the colour value of an ink combination for the second
illumination, we can select an ink combination that has a good match in both
illuminations and thus improve colour constancy.
Following on the above research on improving colour stability across two
illuminations, there are a couple of other approaches used to transfer spectral
data to CIELAB colour space plus an additional 3 dimensions. Nakaya et. al. use
sRGB colour space to represent the remaining 3 dimensions, [89][90]. Derhak
and Rosen introduced LabPQR interim colour space [57], [58]. LabPQR is a 6-
dimensional space where the first 3 dimensions represent the gamut of a printer
in CIELAB under a given illumination. The remaining 3 dimensions represent the
next 3 most important dimensions of the spectral gamut after the CIELAB (or
CIEXYZ) data is removed. The advantage of this interim space is that a hybrid of
traditional gamut mapping and spectral gamut mapping is combined with almost
no additional complexity in comparison to 3-dimensional gamut mapping. Given
an input reflectance, the traditional gamut mapping algorithm can be applied in
CIELAB space. After the mapping, each cell represents a cluster of data points
with the same colour values (CIELAB) but different PQR values, which represent
different spectral reflectances. It was also shown that this interim space can
represent the spectral gamut of a 6-ink printer with very high accuracy [59].
To evaluate the accuracy of the LABpqr method to meet spectral colour
reproduction needs, this method was tested on five different reflectance data
bases. The LABpqr values were calculated based on the 8-ink printer data
75
discussed in Chapter 3: Table 7 shows the round trip result of the model and the
accuracy of the model in embedding spectral characteristics of the five
databases. The performance was evaluated under one reference illumination
(D65) and under 11 different illuminations as specified in Table 6.
The data shows that the model has a small round trip error, similar to what
was reported by Derhak et al. in [57]. This result is consistent with PCA analysis
of the 8-ink printer, which showed that 5 bases were sufficient to represent the
printer spectral gamut accurately (Figure 12). In this experiment, out of the five
databases tested, two included natural reflectances (Simon Fraser University [65]
and Leeds University [82]). The other 3 databases (Leeds, Munsell, MacBeth)
represent printed or painted colours, which typically have lower spatial
complexity than scene reflectances because of the ink limit of the medium and
the number of paints or inks used in the system.
The experiment shows that this model can be sufficient for spectral colour
reproduction of paintings and prints but not for databases that include scene
reflectances with higher than 6-dimensional complexity.
76
Table 7: LABpqr performance of an 8-ink printer system in reconstructing 5 different data sets. Leeds database represents scene reflectances provided by Leeds University [82]. Munsell database includes reflectances for a set of prints and paintings measured by Leeds university. The SFU dataset is provided by Computational Vision Lab at Simon Fraser University [65]. The last two databases include the MacBeth Colour Chart and some pigmented colours used by artists.
Num ∆E94 RMS ∆E2000
Patches mean max max Daylight mean max mean max
max Daylight
8-ink 6048 .18 2.6 .87 .002 .03 0.19 2.43 .91 Leeds 5682 0.96 40.8 7.22 1.77 7.31 0.80 41.32 6.93 Leeds MunSell 719 1.02 28.4 4.2362 1.64 6.92 0.84 41.32 5.82 SFU 1350 0.41 21.7 4.7257 0.02 0.09 0.45 26.47 5.28 MacBeth Colour Chart 24 0.50 7.7 1.3701 0.02 0.05 0.56 9.25 1.38 Pigment 35 0.74 6.7 1.212 0.04 0.06 0.56 9.25 1.38
There were other attempts to enable traditional gamut mapping for
spectral reproduction by lowering the complexity of the gamut space using
methods like PCA [56]. Bakke et al. proposed an improvement to PCA-based
gamut mapping by defining the direction of projection to be towards the centre of
the printer gamut in each 2D cross section of gamut created by a combination
PCA basis and media vector. The 2D cross section is defined by 2 vectors – a
line between the given reflectance and the gamut centre in PCA space – and a
vector representing the spectral gray component of the medium (paper). The
gamut boundary is found by calculating the intersection between the 2-
dimensional plane and the hyper-planes that define the gamut.
77
Spectral Ink Separation Based on Inverting a Printer Model
As discussed earlier there are several methods proposed for modelling the
printer output. Most of these models are based on a non-linear transformation
and, because of that, inverting the models accurately is quite challenging. Di-
Yuan Tzeng et al. [50], [51] divided a 6-ink printer model into several 4-ink printer
models to reduce the complexity of inverting a printer model. The assumption is
that no more than 4 inks are put down on the same location. An iterative search
is used to find the closest match to a given data point in each 4-ink printer model.
Urban et al. introduced a fast method for inverting the Yule-Nielson
cellular Neugebauer model [53][54]. The inversion model is based on a local
search, considering one dimension (one ink) at a time. Given an input
reflectance, this method finds the best ink density of the ith ink given the selected
ink density of the previous inks. His study showed that even though the proposed
search method does not guarantee finding an optimal ink combination, it finds
one that is very close to optimal.
Proposed Method I: Spectral Ink Separation using Non-Negative Least Squares
One method to simplify gamut mapping algorithms is to assume that the
printer gamut is convex in spectral reflectance space and thus there is no need to
have very fine sampling of the printer gamut to capture all of the concavities.
A fast gamut mapping method is introduced in this section that is based on
a Non-Negative Least Square (NNLSQ) method. The Non-Negative Least
78
Square (NNLSQ) method tries to minimize the sum of the residuals using a non-
negative combination of the available data as discussed in the Chapter 1.
Considering the gamut mapping procedure, the goal of the algorithm is to
reproduce an input reflectance that is inside or on the gamut boundary shell and
is closest to the input reflectance. Assuming that the gamut boundary is convex
and relationship between close neighbours can be defined linearly, a point on or
inside a gamut hull can be represented by a linear interpolation of the
neighbouring gamut points. The equation below captures this relationship:
(18)
ρ = Σ αiqi, αi ≥0, Σαi = 1
Where ρ is the input reflectance and qi represents the set of measured
points on or inside the gamut hull. The αi’s are weights, and the restrictions on
the weights ensure that ρ does not lie outside the convex hull of the qi.
For a point ρ outside a convex gamut, we can find the closest point to ρ
lying on the convex hull of the gamut by finding αi minimizing the distance e:
e = | ρ - Σ αiqi |2, αi ≥0, Σαi =1
Finlayson et al. [67] showed that the above equation can be rewritten to
include a weight W as an extra dimension in the input data, and that the revised
equations can then be solved by the standard NNLSQ method. Their derivation is
as follows.
qi′ = [qi W]
79
e′ = | ρ - Σ αiq′i |2
Re-writing e’ yields:
e′ = e+ W*(1 - Σ αi)
The advantage of the above equation is that it can be minimized by
applying NNLSQ. Choosing a large value for W emphasizes the second term in
e′, thereby enforcing the constraint Σαi =1.
Spectral gamut mapping means mapping a spectrum that lies outside the
printer gamut onto a printable spectrum. For a spectrum represented as a point,
ρ, minimizing e′ the closest point on the gamut’s surface, in other words it finds
the closest printable spectrum. The spectrum is described as a linearly weighted
combination of other printable spectra, spectra that are within the printer gamut.
This proposed gamut-mapping algorithm is easy to implement and the
computation is relatively fast considering the dimensionality of the input spectra.
The space and time requirements of the algorithm grow with the number of input
data points. However, as it was shown in Chapter 3:Printer Modelling, by
sampling ink space intelligently, the number of points required to represent a
gamut space can be reduced by as much as 97%.
Preserving Colour for a Desired Illumination
The proposed spectral gamut mapping algorithm using the NNLSQ
method maps an input reflectance to the closest printable spectrum in spectral
space, but there is no guarantee that this new spectrum will have the same
80
colour (for a fixed illuminant) as the original spectrum. It would be preferable to
have a spectral gamut-mapping technique that maps an out-of-gamut spectrum
to the closest in-gamut spectrum subject to the constraint that it preserves colour
under a given illumination as well.
Chau et al. [6] proposed dividing each surface reflectance into two
components. One is the fundamental component that represents perceived
colour under a single illumination and another is the metameric black component
that is invisible to the normal human eye. Using this approach each surface
reflectance, s, can be represented as:
s = fs + bs
where fs represents the fundamental component or basis that represents
colour, and bs represents the metameric black of input surface reflectance. Chau
proposes a gamut mapping algorithm that searches for a reflectance s’ that has
the same fundamental component but may have a different metameric black.
In this section an extension to the NNLSQ algorithm is introduced to
include Chau’s proposed method in order to preserve perceived colour while
finding the closest metameric black for a given input reflectance. This can be
accomplished by modifying the NNLSQ algorithm so that the projection onto the
gamut is in a direction perpendicular to CIEXYZ space under a given illumination.
Doing so preserves the CIEXYZ coordinates as much as possible.
The method works as follows. Let U represent the principal components
basis of the x, y and z colour matching functions. If the visible spectrum is
81
sampled n times, then each x, y
Let P be the set of spectra in the printer gamut {p1, p2, …}, then the
following linear projection of values in P into U
and z basis can be represented as 1xn
dimensional matrix and U will be an nxn matrix, where first row in U captures the
most variance in CIE XYZ colour space and the last row represents the least
variance.
Pu = PU
represents the gamut in U space sorted in terms of decreasing variance in
XYZ space. After this linear transformation, the first 3 coordinates of Pu represent
the tristimulus values of spectra in the printer gamut. Next, weights can be
applied to Pu so that applying the gamut-mapping algorithm described above to
the weighted Pu yields the closest spectrum on the hull of the printer gamut that
creates the least change in CIEXYZ space. The larger the weights are the more
emphasize is put on preserving values in CIEXYZ dimension compared to other
spectral residual variations.
Evaluation of Gamut Convexity
Many existing gamut-mapping algorithms [69], [46] and [44], including the
LabPQR spectral gamut-mapping algorithm [57], map an out-of-gamut point onto
the convex hull of the printer gamut. The assumption is that the gamut is convex.
Is this assumption valid and how much accuracy is lost by assuming a convex
printer space?
82
Algorithms such as Alpha Shape [70] can measure concavity of a space in
a low-dimensional space (3D), but cannot be used in high-dimensional spectral
space.
Comparing the NNLSQ performance to an algorithm that does not depend
on the convex assumption of the gamut can explain how much inaccuracy is
introduced when a printer spectral gamut is assumed to be convex.
Most mappings that do not depend on the gamut convexity assumption
are based on a type of search method. It is proposed that the Hierarchical
Search algorithm introduced in Chapter 3 be used.
Experiment
The printer model used for this experiment was based on the Tzeng (TK)
printer model which was explained in equation 16.
Two types of inks were used, synthetic inks which are square wave based
and sine wave based inks (as discussed in Chapter 2), and inks based on real
pigmented ink reflectance measurements. Three different variations of square
wave inks were used with 0, 10% and 20% overlap in their wavelength
absorption sensitivity region. Similarly, for the sine wave ink, 10% and 20%
overlaps are used.
Four different ink numbers (3, 6, 9 and 12 inks) were used to evaluate the
convexity of the system spectral gamut. For the real ink data, the 3 inks
considered were cyan, magenta, and yellow. For the 6-ink case, the initial 3 inks
were retained, and 3 more complementary inks were added, namely, orange,
83
green, and violet. light cyan, black and light magenta were added to these 6 for
the 9-ink case. For the 12 ink case, gray and medium gray inks along with a light
red ink were added to the system.
Target reflectances were selected as the 80 scene reflectances sampled
from Simon Fraser Database [65] using K-means clustering as explained in
Chapter 3.
For convex hull gamut mapping, the NNLSQ implementation provided by
Matlab was used. The HS search method introduced in Chapter 3 was used to
evaluate concavity of the printer gamut in spectral space. The HS method was
used since the method does not depend on calculating the convex hull of the
printer gamut.
Results
In the previous chapter, it was shown that as the number of primaries
available in the system increases, the overall spectral gamut coverage increases.
The gain in the gamut coverage varies depending on the primary reflectance
characteristics. Figure 44 shows that using the NNLSQ method to map the out-
of-gamut reflectances onto the convex hull of the printer gamut -- similar to the
non-convex mapping method -- the gamut coverage increases as the number of
available primaries increases.
84
Figure 44: Gamut coverage calculated using the non-negative least square technique. The Y axis shows Root Mean Square (RMS) variation between mapped reflectance and the input scene reflectance. SQ0 represents square wave inks with 0% overlap, Sine10 represents sine wave inks with 10% overlap, and Realistic represents real ink reflectance measurements.
To evaluate the concavity of a printer gamut, the variations between the
NNLSQ gamut mapping and HS (Hierarchical Search) gamut mapping were
compared. HS mapping is an iterative search method that does not make any
assumption about the convexity of the gamut, whereas the NNLSQ method
assumes a convex space. Results from the HS method were used as the closest
point on the printer spectral gamut to a given target reflectance. Figure 45
evaluates how different the gamut mapped reflectance using a convex gamut
mapping based on the NNLSQ technique is from a non-convex gamut mapping.
The first observation was that the variation, which indicates the concavity of the
gamut, increases as the number of available inks in the system increases.
Another observation was that square wave inks with a higher degree of overlap
have much larger concavities in their gamuts than inks with less overlap. This is
consistent with the observation from Chapter 3 that inks with a larger than
optimal reflectance sensitivity overlap tend to have smaller gamut coverage. The
85
data in this section shows that most of the change in gamut is in its concavity
rather than the coverage when the inks (more specifically square wave based
inks) have larger overlap than what is optimal.
Figure 46 shows the concavity results for sine wave inks with different
overlap and compares the result against real ink measurements. It shows that
inks with smoother reflectance characteristics (sine wave versus square wave)
have fewer concavities in their gamuts.
Figure 45: Variation between convex gamut mapping and concave gamut mapping. The concave gamut mapping is based on the HS iterative search algorithm and the convex gamut mapping is based on the non-negative least square algorithm. The variation is calculated as average RMS (Root Mean Square) difference between the two spectral reflectances calculated using two different mapping methods.
86
Figure 46: Variation between convex gamut mapping and concave gamut mapping for primaries with smooth variations (sine wave). The concave gamut mapping is based on the HS iterative search algorithm and the convex gamut mapping is based on the non-negative least square algorithm. The variation is calculated as the average RMS (Root Mean Square) difference between the two spectral reflectances calculated using the two different mapping methods.
Perceptual variations between convex and concave gamut mapped
reflectances are presented in Figure 47, which shows the variation in DeltaE94
colour space. The variation was calculated as the average deltaE94 under 11
different illuminations [66].
The first observation was that the concavity assumption of the printer with
sine wave inks has an acceptable variation (1 to 4 deltaE), knowing the gain that
we get by using the NNLSQ technique. Knowing this, the NNLSQ algorithm can
be used as a good candidate for gamut mapping of a non-linear device with a
large number of primaries. The figure also shows that, despite the increase in the
spectral difference between convex and concave gamut mapping as the number
of inks increases, the increase is not perceptually as important. Figure 48 shows
reflectance matches for 6 and 12 ink systems with sine wave inks of 20%
overlap. It shows that in 12 dimensions the convex and concave gamut mapped
87
reflectances have small multiple variations that add up to a large spectral
difference. However, multiple small variations in the spectral space are not
perceptually as visible as a few large local variations in the spectral space.
On the other hand, for the real ink reflectances, there is much higher
perceptual variability when the convex gamut mapping algorithm is used,
compared to what the printer is actually capable of (Figure 47).
Figure 47: Variation between convex gamut mapping and concave gamut mapping for primaries with smooth variations (sine wave). The concave gamut mapping is based on the HS iterative search algorithm and the convex gamut mapping is based on the non-negative least square algorithm. The variation is calculated as average DeltaE94 difference between the two spectral reflectances calculated using two different mapping methods.
88
Figure 48: Reflectance characteristics of HS and NNLSQ gamut mapping for 6 and 12 ink systems. The inks are sine wave inks with 20% overlap.
Proposed Method II: Geodesic Based Ink Separation for Spectral Printing
In this section a new ink separation algorithm is introduced for printing with
6 to 9 inks. A new spectral gamut mapping algorithm is also introduced that
projects an input reflectance onto the manifold of the printer spectral gamut
space. Ink separation, i.e., finding the best ink combination to reproduce a given
reflectance, is done by applying an interpolation between printer gamut points
neighbouring a projected point’s geodesic location. This algorithm was inspired
by the work of McIntosh et al. [71] who suggested interpolating over geodesic
distances, rather than Euclidean distances, to improve an image segmentation
algorithm.
The technique finds the best manifold projection using the ISOMAP
technique explained earlier. The algorithm searches for the lowest dimensionality
that holds the spectral information accurately. This method will aid in finding a
good ink combination given an input reflectance for 6-ink and 9-ink printer
models.
89
Use of Thin Plate Spline Interpolation in Spectral Reproduction
For the spectral printing process, it is proposed that TPS be used to find a
continuous function that maps between the set of inks and each of the output
dimensions. For instance, if the output spectral reflectance of an 8-ink printer is
measured from 380nm to 730nm with a 10 nm sampling, TPS is used to create
36 separate functions mapping from the 8 input dimensions to each reflectance
wavelength.
Geodesic Interpolation and Ink Separation
Interpolation is a common approach to ink separation and the ink
separation technique introduced in this section is also based on interpolation. In
general, an ink combination is interpolated as a weighted combination of nearby
experimentally measured data points. The weights typically are derived based on
the distance of the point to be interpolated from its neighbours. The distance
metric can be defined in many different ways. For instance, the distance between
two spectral reflectances can be measured as the Euclidean distance between
them. In this section, an interpolation-based ink separation algorithm based on
geodesic distances over the gamut manifold is proposed.
Many spaces appear to have a high dimensionality in a linear space, but
actually have lower intrinsic dimensionality. A good example is the Swiss Roll
example discussed in Chapter 2.
The proposed ink separation method introduced in this chapter uses the
geodesic distances between data points. The algorithm is as follows:
90
1. Given a set of training points (reflectances of print samples), the
geodesic distances between the input reflectance (the reflectance to be printed)
and all training points in the gamut are calculated
2. The geodesic distances are used in an MDS engine (Multi-
Dimensional Scaling) to calculate the point locations in a space of lower
dimension.
3. Thin Plate Spline interpolation is used based on the data point
locations in the new space
3a. Weights for the interpolation are calculated based on the
distance of the point from the neighbouring points in the lower dimensional
space.
Steps 1 and 2 are part of the standard ISOMAP algorithm [62]. ISOMAP
makes the assumption that the Euclidean distances to points within the local
neighbourhood of a given point, P, approximate the corresponding geodesic
distances. The geodesic distance to a point, Q, outside the local neighbourhood
is calculated as the sum of the distances between neighbouring points along the
shortest path from P to Q.
Spectral Gamut Mapping based on Manifold Projection
In this section, a possible spectral gamut mapping algorithm based on
manifold projection is presented. Figure 49 represents an example where gamut
mapping in Euclidean space may not result in the closest point in the gamut.
91
However, mapping based on the geodesic location of the gamut points can result
in a closer (true) mapped location (Figure 50).
The proposed mapping algorithm has the following steps:
1. Given a printer gamut space, calculate the data point’s geodesic
location using ISOMAP
2. Transfer the input spectral reflectance using the same
transformation
3. After the transformation, gamut mapping is applied in the (lower
dimensional) transformed space
a. The Non-Negative Least Square gamut mapping algorithm
introduced in the previous section is used for mapping the
transformed input reflectance onto the printer gamut
4. The projected value in the lower dimensional space is inverted back
to the spectral space using an interpolation method
a. Thin Plate Spline interpolation is used for the inverse
transformation
Another advantage of applying gamut mapping based on the manifold
projection is that the mapping can be done in a much lower dimensional space.
Because of reduce dimensionality, the time and space complexity of the gamut
mapping can be reduced significantly.
92
Figure 49: Example of the gamut mapping algorithm in Euclidean Space. Blue lines (Swiss roll) represent a device gamut and the green point represents an out of gamut point.
93
Figure 50: Gamut mapping method using ISOMAP where the projections are applied in a lower dimensional space.
Evaluation Method
Similar to the previous gamut mapping algorithm (NNLSQ), the accuracy
of the manifold projection algorithm is evaluated by comparing the result of the
mapping algorithm against what the Hierarchical Search (HS) mapping finds.
Details of HS gamut mapping are explained in Chapter 3.
Performance of the manifold projection method was evaluated by
comparing the RMS (Root mean square) differences of the mapped reflectance
using this technique with the HS search method results. In addition to RMS,
average deltaE94 colour difference of the mapped reflectances using the
proposed method and HS search method under 11 different illuminations is
presented.
94
To evaluate the difference between the projected reflectance and the
results from the HS search algorithm the manifold projected value needs to be
projected back to the spectral colour space. Thin Plate Spline interpolation, which
is defined in the previous chapter, is used for inverting the ISOMAP
transformation.
Time and Space Complexity
There are several methods to calculate Geodesic distances between
points given the distances between neighbouring points. Most commonly,
Dijkstra’s algorithm is used to find the shortest path between each point in the
data set [2]. If there are Mp points representing the printer gamut, and MD input
points for ink separation, the time complexity of Dijkstra’s algorithm based on the
Fibonacci heap algorithm is O(E + (Mp+MD)Log(Mp+MD)), where E represents the
number of edges between the points. The number of edges varies with the data
set characteristics and diameter of the neighbourhood around each data point. In
practice, it takes around 2.5 seconds to calculate the geodesic distances for
2000 points on an average computer.
Experiments
Printer Gamut
To evaluate the gamut complexity of the printer, two printer gamuts were
evaluated. The first one was based on spectral measurements of 1350 patches
printed with an 8-ink printer. This gamut is referred to as a “realistic” printer
gamut since the measurements were based on the real printer gamut
95
measurements. The second printer gamut was based on the synthetic printer
model introduced by Tzeng explained in Chapter 3. The model was used to
predict the spectral reflectance resulting from a given ink combination.
Ink Choices
Two sets of ink selections were used in this study. The first set was based
on real ink measurements as discussed in previous chapters. The inks were
orange (O), cyan (c), magenta (m), yellow (y), green (Gr), violet (V) and black
(K), light magenta (LM) and light cyan (LC) for the 9-ink printer. Orange (O), cyan
(c), magenta (m), yellow (y), green (Gr) and violet (V) were used for the 6-ink
printer, and the 3-ink printer model used cyan (c), magenta (m) and yellow (y).
The second set of inks had synthetic ink reflectances with a square wave
shape and 0% overlap in their absorptions sensitivity region.
Printer Spectral Gamut Intrinsic Dimensionality
What are the intrinsic dimensionalities of the gamuts of the two printers?
In terms of a linear model, Principal Component Analysis (PCA) provides one
answer. However, in terms of a non-linear model, ISOMAP provides a second
answer. If the answers differ, then it can be conclude that the printer gamuts
bend in a way that is analogous to the Swiss Roll example. Figure 51 and Figure
52 compare how the residual variance changes with increasing dimensionality for
both PCA and ISOMAP. If PCA shows a higher dimensionality for the data set
than what ISOMAP finds, then we can conclude that the underlying structure of
96
the gamut is a lower-dimensional data set. Figure 51 and Figure 52 are based on
the realistic gamut; Figure 53 and Figure 54 are for the synthetic gamut.
The analysis shows that the printer gamuts are of lower (3 or 4 versus 5 or
6) intrinsic dimensionality than can be determined by linear PCA. In addition, the
weights used for the interpolation based on the new proposed method are
calculated using geodesic distances of the points in lower dimension. As a result,
it should be possible to obtain more accurate ink separations using interpolation
based on the distances between the ISOMAP-embedded locations of the
reflectances.
Figure 51: PCA residual variance for the realistic printer gamut space. The plot shows that the dimensionality of the 7-ink printer is around 5 dimensions.
97
Figure 52: ISOMAP residual for the realistic gamut. The data shows that the underlying dimensionality of the gamut is around 3.
Figure 53: PCA residual variance for the synthetic printer gamut space. The scores show that the dimensionality of the 6-ink printer is around 6 or 7 dimensions.
98
Figure 54: ISOMAP residual for the synthetic gamut. The data shows that the underlying dimensionality of the gamut is around 3 to 4.
Results
To evaluate how well the ink separation technique works, the synthetic
printer gamut was sampled uniformly in ink space, obtaining 2300 data points as
a training data set. An additional 250 data points from inside the printer gamut
were selected to represent the test sample. The test and training sets were
disjoint.
The 250 test points were processed through the ink separation algorithm.
The predicted ink combinations were then run through the printer model to
predict the corresponding spectral reflectances. The predicted reflectances were
then compared to the original input reflectances.
To evaluate the performance of the geodesic ink separation model, its
results were compared to those obtained by doing the separation in linear space.
Table 8 shows that there is a gain when the interpolation is based on the
geodesic distances instead of the Euclidean distances.
99
Table 8: Ink separation evaluation based on geodesic location and linear space locations. The errors reported are the minimum, mean, and max ∆E94 that occur under the 11 different illuminations, and the RMS difference between the spectra.
Geodesic Linear Inks RMS ∆E94 RMS ∆E94
3
min 0.0011 0.236 0.0078 1.204 mean 0.034 3.068 0.0335 6.33 max 0.134 8.2 0.0717 24.78
6
min 0 0.005 0 0.006 mean 0.0089 2.843 0.0298 3.379 max 0.0541 18.29 0.1339 20.45
9
min 0 0.004 0 0.014 mean 0.0081 2.617 0.0179 3.051 max 0.0487 14.1 0.1238 20.05
Figure 55: Ink separation methods evaluated in ∆E94 under 11 different illuminations. The data above shows average ∆E94 for the 11 illuminations
Spectral Gamut Mapping Evaluation
Test Data
To test the gamut mapping algorithm, the scene reflectances from the
SFU database were used [65]. There are 1350 individual reflectances in the
100
database. Figure 56 and Figure 57 show the accuracy of the spectral
reproduction when the proposed spectral gamut mapping algorithm is used
compared to the hierarchical search method to map the out-of-gamut points onto
the gamut hull. The table shows that the proposed gamut mapping algorithm is
as accurate as, or better than, the hierarchical search algorithm.
It is important to keep in mind that the hierarchal search algorithm has
some inaccuracy as well which can add to the overall evaluation comparison.
Some of the inaccuracy of the hierarchical search algorithm comes from the
sampling resolution of each ink axis. The higher the sampling resolution, the
more accurate the model. For these experiments, the hierarchical search had 6
levels, and at each level the sampling resolution for each axis was 5. For
instance, for a 3-ink system, at each search level there are 53 different ink
combinations to choose from. Once the closest ink combination was selected
(Pa), 53 samples were selected close to the point Pa. This process was repeated
for 6 levels.
101
Figure 56: Comparison between ISOMAP-based gamut mapping and HS gamut mapping for realistic and synthetic ink reflectances. The variation is calculated as the average RMS difference between the two spectral reflectances calculated using the two different mapping methods.
Figure 57: Comparison between ISOMAP-based gamut mapping and HS gamut mapping for realistic and synthetic ink reflectances. The variation is calculated as average DeltaE94 difference between the two spectral reflectances calculated using the two different mapping methods.
Figure 58 compares accuracy of gamut mapping based on Isomap
technique versus NNLSQ (non-linear least square) method. Two types of inks
102
were used in the comparison: square wave inks with no overlap and real ink
measurements. Figure 59 and Figure 60 compare performance of the two
mapping algorithm in DeltaE94 colour space. The two figures show that NNLSQ
gamut mapping has similar performance to the ISOMAP method. This means
that, not much of printer gamut concavity is reduced by transforming the printer
gamut using ISOMAP technique.
Figure 58: Comparison between accuracy of ISOMAP-based gamut mapping and NNLSQ gamut mapping measured in mean RMS. Accuracy is defined as variation between how the gamut mapping performs compared to HS gamut mapping
103
Figure 59: Comparison between accuracy of ISOMAP-based gamut mapping and NNLSQ gamut mapping measured in mean DeltaE94 for square wave ink with 0% overlap. Accuracy is defined as variation between how the gamut mapping performs compared to HS gamut mapping
Figure 60: Comparison between accuracy of ISOMAP-based gamut mapping and NNLSQ gamut mapping measured in mean DeltaE94 for real inks. Accuracy is defined as variation between how the gamut mapping performs compared to HS gamut mapping
104
Conclusion
A spectral ink separation algorithm is introduced based on interpolation
using the geodesic distances between neighbouring points. A spectral gamut
mapping algorithm is also introduced which uses ISOMAP.
The performance of the ink-separation model was evaluated for 3-ink, 6-
ink and 9-ink printers using a synthetic printer model. The experimental results
show that the accuracy of interpolation, and thus of the resulting ink separation,
improves if the calculation is done using geodesic distances.
In addition, a new spectral gamut mapping algorithm is introduced based
on manifold transformation of the printer gamut.
105
CHAPTER 6: EVALUATION OF SPECTRAL COLOUR REPRODUCTION
Introduction
In this section, an experiment with real inks is presented to evaluate how
closely a sample colour can be reproduced using available real inks, and whether
the accuracy of spectral reproduction is noticeably better than traditional tri-
chromatic colour reproduction.
Target Samples
Two sets of target samples were used. The first set was based on the
MacBeth colour checker shown in Figure 61. To have a more focused evaluation
of spectral reproduction given a limited set of inks, 3 colour tiles with similar
reflectance characteristics were used in the experiment (as shown in Figure 61).
Figure 62 shows the reflectance of the 3 colour patches and Table 9 represents
the colour variation for each patch under 11 different illuminations from the
Simon Fraser database.
106
Figure 61: The 3 colour tiles used from the MacBeth Colour Checker for spectral colour reproduction testing. The colour tiles used were the 5th, 8th and 13th colour tiles as indicated by the red mark.
107
Figure 62: Reflectance characteristics of the 3 colour patches selected from the MacBeth Colour checker as target reflectances for reproduction.
Table 9: Average Colour variation (inconsistency) of each patch under 11 different illuminations. Mean DeltaE94 column represents average colour variation from target patch
Patch Number Mean
DeltaE94
5 2.1 8 3.2 13 5.8
The next set of test targets was based on real paint samples that are
specifically hard to reproduce. For the purpose of this study, 2 yellow paint
samples were selected with the reflectance and colour inconsistency variations
shown in Figure 60 and Table 10.
108
Figure 63: Reflectance characteristics of the 3 yellow paint samples selected as targets. Table 10: Colour variation (inconsistency) of the 2 yellow paint patches under 11 different illuminations.
Patch Number DeltaE94
S_Y1 3.7 S_Y2 6.6
Experiment Setup
A 9-ink printer with 3 different magenta inks (Figure 64), 2 red inks (Figure
65) and 2 different yellow inks (Figure 66) plus a cyan and a black ink was setup
to reproduce the selected target samples. The selection of the 9 inks is based on
limited set of available inks during the experiment and also the colour range of
the target samples. The printed patches were optimized for typical lighting
condition found in viewing light booths as shown in Table 11.
109
Figure 64: Reflectance characteristics of the 3 different magenta inks used for the experiment.
Figure 65: Reflectance characteristics of the 2 different red inks used for the experiment.
110
Figure 66 Reflectance characteristics of the 2 different yellow inks used for the experiment.
Table 11: 4 Light sources available in the viewing light booth
Light Sources Daylight D65 Incandescent light A Cool White Fluorescent CWF Department store light - TL 84
Implementation Details
To evaluate spectral colour reproduction, each target colour patch was
printed using spectral printing and traditional colour reproduction which matches
a colour in CIELAB colour space (trichromatic matching).
For CIELAB colour reproduction, an illumination was selected from the
available 4 choices that had the most colour variability from the other 3. This was
done to magnify the effect of spectral printing given the limited number of
available inks. Because of this, the same light source was not used for the entire
target under trichromatic (CIELAB) colour reproduction.
111
The printer modeling used for the experiment was based on the modified
YNCN model explained in Chapter 2. To reduce the number of training patches
needed for the application, the printer gamut was divided into two parts; one part
focuses on the yellow part of the printer gamut and the other focuses on the blue
region.
For trichromatic colour reproduction, a convex-hull gamut mapping was
implemented to map out-of-gamut points onto the printer gamut. For the ink
separation process, the Hierarchical Search algorithm was implemented to find
the best ink combination given the mapped input reflectance.
Since the ISOMAP technique has a better performance than NNLSQ
technique for gamut mapping, the gamut mapping and ink separation algorithms
are both based on the ISOMAP technique (explained in Chapter 4).
Results
For the MacBeth Colour Checker samples, patches number 5 and 8 were
matched under Cool White Florescent light. Patch number 13 was matched
under Daylight illumination. Figure 67, Figure 68 and Figure 69 show the
accuracy of colour reproduction under CIELAB colour space and spectral space
for the 3 selected MacBeth colour patches.
112
Figure 67: Accuracy of reproduction of patch number 5 under 4 different illuminations using trichromatic matching versus spectral colour reproduction. [84]
Figure 68: Accuracy of reproduction of patch number 13 under 4 different illuminations using trichromatic matching versus spectral colour reproduction.
113
Figure 69: Accuracy of reproduction of patch number 13 under 4 different illuminations using trichromatic matching versus spectral colour reproduction.
For the yellow patch, S_Y1, given the available two yellow inks, spectral
reproduction could not find an ink combination that has better metamerism than
that of trichromatic matching.
For S_Y2, an illumination was selected as the reference lighting for the
trichromatic matching. Figure 70 shows the accuracy of each reproduction
process measured as DeltaE94 colour variation. The figure shows that when
trichromatic matching is used, the selected ink combination can reproduce a
closer result to the target patch under one illumination. However, spectral colour
reproduction on average produced closer reproduction than trichromatic
matching.
114
Figure 70: Accuracy of reproduction of the yellow patch S_Y2 under 4 different illuminations using trichromatic matching versus spectral colour reproduction. The trichromatic matching was done under Incandescent light A illumination.
115
CHAPTER 7: SPECTRAL ANALYSIS OF BRONZING
Introduction
Until now the focus of this study has been on improving metamerism as
one of the main advantages of spectral reproduction. In this section, another use
of spectral colour reproduction, understanding and optimizing Bronzing, is
discussed. Bronzing has recently been identified as one of the important factors
in improving the colour reproduction process.
Bronzing is caused by reflection of light from the ink when the printed
image is viewed at a particular angle. Bronzing is pronounced with pigmented
inkjet inks because the pigments do not penetrate into the ink-receiving layer of
the print medium. Rather, the pigments form a film or layer on top of the ink-
receiving layer. One common type of bronzing is caused by interference between
the light reflected from the top surface of the inkjet film and the light reflected
from the interface between the inkjet film and the ink-receiving layer. This type of
bronzing is typically observed with black-pigmented inkjet inks and varies with
the thickness of the inkjet medium coating. Therefore, bronzing is more
pronounced on glossy, photo-based print media than on paper-based print media
[72].
To reduce bronzing, different colour mappings (dot placement algorithms)
[72] and different ink selections are used. Some works have focused on use of
composite colours instead of primary colours [73] or adding a gloss optimizer to
116
the printer system. Currently, most methods available for measuring and
evaluating bronzing are based on an subjective evaluation where engineers
compare reflectance variations of prints under different viewing angles.
The focus of this section is to show another application of spectral
analysis aimed at improving the colour reproduction process, something that is
not feasible using traditional tri-chromatic analysis. The proposed algorithms try
to address the issue of detecting and measuring the bronzing defect, knowing
that, at this time, there is no subjective metric for measuring bronzing. The
proposed models are based on automating measurement of the bronzing. A new
metric to evaluate this defect is also proposed.
In the first part of this section, a new method of understanding and
measuring bronzing is proposed based on variations in spectral reflectance of an
ink under a range of viewing angles. In the second part of this section, a
modeling method is proposed to predict the reflectance of a given ink under
certain angles. This method permits the evaluation of bronzing without the need
to measure a print under a large range of viewing angles. This algorithm then is
extended to model bronzing of an ink at different densities.
Using this technology, a subjective metric to evaluate bronzing can be
designed. The second advantage of this algorithm is that industries can quickly
evaluate how the bronzing profile of an ink changes under different densities of
the given ink.
117
Data Measurement
For bronzing evaluation, a Gonio-spectrophotometer was used to measure
spectral reflectance of a patch under different viewing angles in 2 dimensions. An
initial calibration against a perfect diffuse white was performed.
Based on precise control on the incident and acceptance angle by the
computer, a Gonio Spectrophotometer measures the spectral distribution of
every pair of illumination/detection angles individually. After calibration and
mounting of the samples, the settings for measurement conditions are made.
Bronzing Evaluation based on Spectral Reflectance Characteristics
In this section a method for evaluating bronzing based on the spectral
reflectance variation of a printed plot under different viewing angles (assuming
the illumination incident angle is constant) is introduced. Figure 71 shows the
spectral reflectance of a black ink patch at viewing angles of 10 to 15 degrees in
steps of 0.5 degrees. The incident angle was kept constant at 20 degrees. The
prints were made on photo paper.
The photo paper medium has a coating that scatters light much less than
typical paper. Since the energy reflected from a photo medium increases at its
specular angle, where incident angle equals viewing angle, it is necessary to
remove any artefacts that are due to shininess of the media, which do not
represent ink and media colour variation. One proposal is to normalize the
measurement data by the measurement of the white of the media at the same
angle (Figure 72). The normalization process divides the spectral measurement
118
of a patch by the reflectance measurement of the same paper (medium) without
any ink on it. For normalization, the white medium was measured at the same
angle as the printed patch.
Figure 71: Spectral Reflectance Variation of black ink (K1) as viewing angle changes (keeping incident angle constant).
119
Figure 72: Spectral reflectance variation of the black ink (K1) under different viewing angles normalized by reflectance of the white paper under the same angles.
Another observation is that reflectance characteristics of inks are a
function of the difference between the viewing and incident angles in 2
dimensions. Figure 73 shows this difference, which is typically referred to as
phase angle, is highly correlated to the reflectance variation. Through the rest of
this paper, phase angle rather than incident or viewing angle is used as the main
parameter for evaluation reflectance variation of a target.
120
Figure 73 Reflectance of K1 ink measured at two different viewing angles. The data is plotted against the difference between the viewing and incident angles.
Through a visual evaluation of plots and their normalized reflectance
variation under different phase angles, a metric is proposed to evaluate bronzing
based on reflectance variation characteristics of a target. The theory behind the
metric is that the plots related to higher bronzing levels will have larger, more
non-uniform spectral variation at different phase angles. Figure 74 shows a type
of black ink that tends to have a large degree of bronzing at high densities and
not much bronzing when little ink is put down.
The proposed bronzing metric is based on variations in spectral
reflectance of the target under different viewing angles as given in equation (19).
(19)
mediaWhiteinknormalized SpectraSpectraSpectra /=
∑=
n
inormalizediSpectraRange
1, )(
121
where Range calculates the maximum value minus minimum value of the
reflectance taken over all wavelengths.
Figure 74: Maximum reflectance variation of an ink under two different densities when considered under a set of viewing angles (keeping incident angle constant). The selected set of viewing angles are+- 10 degrees of incident angle.
Results
In the experiments, the 5 different inks shown in Figure 75 were used.
Some inks, such as magenta and K3, are known to have little bronzing, whereas,
the others do show a significant level of bronzing. Since inks are known to have
different behaviours for bronzing under different densities, two density levels
were considered in the experiment (Low and High densities).
122
Figure 75: Spectral characteristics of the 5 inks used in the study. K1, K2 and K3 represent 3 different types of black ink tested. C represents cyan ink and M represents magenta ink.
Each patch was printed and measured using a Goni-Spectrophotometer at
a 20 degree incident angle. The viewing angle was changed from 10 degrees to
30 degrees in 0.5 degree increments. All the measurements were captured in
spectral colour space and normalized by the white reflectance of the medium at
the same viewing and incident angles.
Figure 76 shows the bronzing metric calculated based on the proposed
method. The figure shows that using the proposed metric, K1 and K2 inks have a
large level of bronzing at different densities which agrees with what is generally
known about the bronzing characteristics of the two inks.
123
Figure 76: Bronzing level calculated based on the proposed bronzing metric. K1, K2 and K3 represent 3 different types of black ink tested. C represents cyan ink and M represents magenta ink.
Modeling Bronze
In this section, the existing printer output modelling techniques are
extended to model other quality metrics such as bronzing.
Predicting Reflectance under Different Viewing Angles
Figure 77 shows the reflectance variation of a given ink at the example
wavelength of 480nm as the viewing angle changes, while the incident angle is
held constant. The figure shows that an interpolation technique can be used to
predict reflectances under different incident angles. The proposed method
interpolates between measurements of reflectances under different incident
angles in order to predict a patch’s reflectance as a function of viewing angle.
The equation below captures the process:
124
(20)
,....),,( 3,2,1,, θλθλθλθλ RRRFR =
where F represents the continuous function created using an interpolation
method and 1,θλR represents the measured reflectance at wavelengthλ and
angle ѳ1.
Figure 77: Variation of the black ink (K1) reflectance at 480 nm wavelength under different viewing angles. The incident angle was kept constant at 20 degrees.
Using this model to predict the reflectance variation of an ink under any
phase angle, a process to measure bronzing with much fewer input data points
can be created.
125
Modelling Bronze for Different Ink Densities and Phase Angles
Printer output models such as the Yule-Nielson Cellular Neugebauer
model or the ISOMAP based model try to predict reflectance of a printed patch
by running an interpolation method for each of the wavelengths. In this section,
the printer output modelling algorithms are extended to predict the reflectance
characteristics of an ink under different phase angles for different densities.
In the previous section, it was shown that using an interpolation method, it
is possible to predict the reflectance of an ink under any phase angle for a given
ink density. The problem that is addressed in this section is whether the
interpolation can be further extended to predict the reflectance of an ink under
any phase angle for any ink density.
Figure 78 shows reflectance variation of a black ink at 3 different ink
densities. It shows that, considering each phase angle for each wavelength, the
spectral variation of an ink as density varies can be predicted.
Based on this observation, a more sophisticated model can be created by
combining the printer modelling algorithms introduced in Chapter 2 with an
interpolation to predict reflectance variation of fixed ink density under different
phase angles. This new method can enable the researcher to predict the
bronzing amount of an ink for different ink densities.
126
Figure 78: Reflectance variation of 3 different ink densities at 480 nm wavelength.
The equation below captures the proposed algorithm for measuring bronzing for different ink densities, where F is an interpolation function, ѳ is the phase angle, λ wavelength, and D is the density of the ink being studied.
(21)
,....),,( 3,,2,,1,,,, DDDD RRRFR θλθλθλθλ =
What can be learned from this model?
One of the common techniques for reducing bronzing is to adjust the ink
separation table within the printer to optimize the ink combination, knowing that
some inks have less bronzing at specific ink densities. For instance, consider a
printer system that has cyan, magenta, light cyan, light magenta, yellow, light
gray and black inks available. In this system, the medium gray ink may have the
127
lowest bronzing for having a neutral gray colour at 20 CIE L*; whereas, for darker
regions using a composite ink, (cyan+magenta+yellow) might reduce the
bronzing significantly.
Knowing this information, the ink separation table can be modified to
minimize bronzing, which is very similar to the process for reducing metamerism.
Experiment Setup
Two different black inks (K1 and K2) were measured. For training
purposes, 6 different densities for each ink were printed. The printed patches
were measured at -10,-8,-4,...10 phase angles. Figure 79 shows the reflectances
of the two inks used in the study. The two black inks have a high level of
bronzing.
Five ink densities, not including the training ink densities, were used for
the test purposes and their spectral reflectance was predicted at -10 to 10 phase
angles with steps of 0.5 degrees. Since the reason for designing this modelling
algorithm is to be able to predict reflectances accurately enough in order to better
understand bronzing behaviour of the patches, the Bronzing Metric, introduced in
the previous chapter, was used for comparison purposes. For interpolation,
weighted Locally Linear Interpolation (LLI) was used and, for printer modelling,
the Yule-Nielson Neugebauer model was selected.
128
Figure 79: Spectral characteristics of the 2 black inks used in this study. The measurements were collected using an eye-1 spectrophotometer at 2 degrees observer angle.
Results
Using the proposed model, the spectral reflectance of each patch was
predicted under 40 different phase angles (-10 degrees to 10 with increments of
0.5 degrees). The proposed bronzing metric introduced in the previous chapter
was used to predict the bronzing metric. The predicted bronzing metric was
compared to a calculated bronzing metric based on real measurements in Figure
80. Equation 19 combined with 21 was used to calculated bronzing. The figure
shows that the modelling algorithm can be used to decide which inks, at which
densities, have lower or higher levels of bronzing.
129
Figure 80: Performance of the bronzing model compared to real measurements. The vertical axis represents the bronzing metric that was proposed in the previous section. D1 to D5 show the 5 different densities used for each ink density. D1 is lightest ink density and D5 is the highest ink density. K1 and K2 are the 2 black inks used.
130
CHAPTER 8: SUMMARY
When primaries interact with each other non-linearly as in a printer
system, modelling the system output becomes complicated. The Yule-Nielson
Cellular Neugebauer (YNCN) model is known to have the highest accuracy in
predicting any ink combination relative to other existing models. The drawback of
YNCN is that it requires significantly more measurement points than other
models. In this thesis, two models were proposed for improving the YNCN model.
The first one takes advantage of the paper (medium) constraints, such as ink
limiting, to reduce the necessary number of measurements substantially (by as
much as 97%). A better, smarter sampling method based on the linearization
curves can relax the required number of measurement points even further. The
second proposed model is based on using a non-linear transformation, ISOMAP,
to reduce the complexity of the output system before any interpolation is applied.
Two conference papers are published for these proposed methods ([68], [78]).
Finding the optimal spectral characteristics for each primary and the
number of primaries needed for a given system is another focus of spectral
reproduction research. Hardeberg [30] showed that in a real system, due to the
noise associated with each primary, the accuracy of the system does not always
improve when more primaries are added.
The effect of different system behaviours (linear or non-linear) and the
reflectance characteristics of the primaries on the accuracy of spectral colour
131
reproduction were evaluated. In particular, the spectral characteristics of each
primary and the number of primaries to use were both studied. The results will be
useful to a researcher in deciding how an existing output device (printer,
projector, scanner or a camera) might be improved by introducing an additional
light source or ink with specific reflectance or absorption characteristics. Another
important finding of the study was the effect of overlap between reflectance
characteristics and their smoothness on spectral colour reproduction accuracy.
Two models, one linear and one non-linear, were used and a hierarchical search
algorithm was implemented to evaluate the system. The results are published in
NIP23 paper [76] and NIP25 [85] .
The next topic addressed in the thesis is that of an algorithm to
determine the best primary combination to reproduce a given spectrum. The
majority of the mapping algorithms in this field are based on calculating the
convex hull of the gamut, and using the hull to determine if an input spectrum is
out of gamut. To reduce the complexity of these algorithms when applied for
spectral analysis, some researchers have proposed using a lower-dimensional
space to make spectral gamut mapping more manageable. Other methods
attempt to invert forward device models, but these tend to have high time
complexity and low accuracy.
Two gamut mapping and two ink separation methods are proposed in this
thesis. The first gamut mapping algorithm proposes a new approach to calculate
the convex hull for only the relevant portion of the gamut space. This model also
calculates the best ink separation assuming a convex output gamut. The main
132
advantage of this algorithm is that the complexity of the algorithm grows linearly
with the number of available points in the system, which is a great improvement
compared to the other convex hull based approaches in which their time
complexities grow at a polynomial rate. The results of the work are published in
the CIC 2006 conference [77].
The second gamut mapping algorithm is based on mapping the gamut in a
low-dimensional space using a non-linear transformation before any known
gamut mappings are applied. There are two main advantages to this method.
First, the model maps the printer spectral gamut space to a lower dimensional
space more accurately than other linear approaches (e.g. PCA or ICA). Second,
after the mapping the spectral gamut space has fewer concavities compared to
linear dimensional reduction methods.
The second proposed ink separation method is based on multi-
dimensional interpolation from the input spectral gamut to output ink combination.
This interpolation approach uses geodesic distances to calculate weights used in
the interpolation instead of standard Euclidean distances [79].
Also, real experimental results were used to evaluate the performance of
the proposed model against one of the conventional tri-chromatic printing
models. The setup has a printer with 9 inks with different colours, some with
similar hue angle: magenta1, magenta2, magenta3, Red1, Red2, yellow1,
yellow2, cyan and black. The experiment is aimed to reproduce the colour of
some tiles from a Macbeth Colour Checker, and some yellow painting patches as
accurately as possible in spectral space. The forward printer model used is
133
based on a modified Yule Nielson Neugebauer model. The spectral gamut
mapping algorithm is based on the proposed mapping after a non-linear
transformation. The ink separation algorithm is also based on the proposed
technique of using Geodesic distances in the interpolation process.
In the last section of this study, we model the reflectance variation of a
printed patch under different phase angles. A metric to evaluate bronzing is
proposed based on the reflectance variation. The metric is based on the spectral
reflectance variation of a patch under different phase angles. This can be a great
advancement in measuring bronzing where there is no simple objective method
exists at this time. Lastly, a method to predict how bronzing varies as a function
of ink density is proposed. It combines the proposed bronzing measurement with
existing spectral printer modelling. Knowing this model, the spectral colour
reproduction process can be improved to consider both metamerism and
bronzing effects.
Detailed Contributions
Two spectral modelling algorithms for printer output were introduced. The
first algorithm is based on optimizing the Cellular Neugebauer Model, which
takes advantage of ink limiting and linearization information to minimize the
number of ink combinations to measure. A modification to the model is proposed
to handle missing measurement points due to ink limiting. The second spectral
modeling method is based on using the ISOMAP technique to map the printer
gamut into a space where the primaries have less non-linear interaction between
one other.
134
The effects of absorption or reflectance sensitivity for each type of primary
and the number of primaries on the accuracy of spectral colour reproduction
were evaluated. A study was performed for output devices with both linear and
non-linear interactions between the primaries. Four different types of synthetic
primaries and three different overlap amounts for each primary type were used in
the study. Finally, the effectiveness of the existing inks was compared against
the optimized synthetic inks for spectral colour reproduction. The results from this
study helps engineers, when designing inks, filters or other types of primaries, to
improve the existing spectral reproduction accuracy of their output device,
whether the device is a projector or a printer. The study also helps engineers to
better understand the gain in spectral gamut of a device if an additional primary
is added to the system.
Since spectral gamut mapping is an important part of the spectral colour
reproduction process and the mapping algorithm is more complex for non-linear
output devices, two spectral gamut mapping algorithms were introduced for
printers. One of the algorithms is based on calculating the convex hull for only
regions of interest in a printer gamut. The proposed method has a time and
space complexity that grows linearly with the number of data points representing
the printer gamut. The algorithm is also used to evaluate the concavity of the
printer gamut in spectral space as the number of primaries increases.
The second spectral gamut mapping algorithm is based on gamut
mapping in a lower dimension space as calculated using ISOMAP. The accuracy
of the proposed gamut mapping algorithm is comparable to running an
135
exhaustive search in the printer gamut. In addition, an ink separation algorithm
was proposed based on interpolation using Geodesic distances between gamut
points. Both of the proposed algorithms are flexible and can handle printer
gamuts with different level of complexities. This is the main advantage of these
algorithms over existing techniques such as LABpqr where the complexity of the
device output gamut is assumed not to be larger than 5 or 6 dimensions.
To evaluate the feasibility of the proposed methods using the existing
printers and inks, an actual spectral colour reproduction was compared against
colour reproduction in CIELAB colour space for a real 9-ink printer system. The
results showed that, given the limited number of available inks, using spectral
colour reproduction method, metamerism could be reduced by around 40%.
Lastly, another application of analysis in spectral colour space was
presented. It was shown that using spectral analysis of a printed patch, a
bronzing metric can be defined. The metric can also be extended to assist
engineers to minimize bronzing in colour reproduction.
136
REFERENCES
[1] R. S., Burns, “Methods for Characterizing CRT displays”, Displays, Volume 16, Issue 4, 1996; 173-182
[2] Tamura N, Tsumura N, Miyake Y. “Masking Model for Accurate Colorimetric
Characterization of LCD”. Proc. IS&T/SID 10th Color Imaging Conference 2002; 312-316.
[3] D., Wyble, M., Rosen, “Color Management of DLP Projectors”, Proc. IS&T/SID 12th Color Imaging Conference 2004, 228-232
[4] B.V. Funt, B. Bastani, X R. Ghaffari, “Optimal Linear RGB-to-XYZ Mapping for Color Display Calibration”, Proc. IS&T/SID 12th Color Imaging Conference 2004; 223-227
[5] Xrite, < http://www.xrite.com/documents/literature/gmb/en/spectrolino_serial_5_en.pdf >, Xrite, Spectralino SpectraScan, 2009
[6] W. Chau, and W.B. Cowan, “Gamut Mapping Based on the Fundamental Components of Reflective Image Specifications”, Proceedings of 4th IS&T/SID Color Imaging Conference, 67-70.
[7] Ian E. Bell, Ian Bell Consulting, Simon K. Alexander, “A Spectral Gamut-Mapping Environment with Rendering Parameter Feedback”, Eurographics, 2004
[8] Arne M. Bakke, Ivar Farup, and Jon Y. Hardeberg , “Multispectral gamut mapping and visualization: a first attempt”, SPIE, Color Imaging X: Processing, Hardcopy, and Applications, Reiner Eschbach, Gabriel G. Marcu, Editors, 193-200, 2005
[9] L. Yang and S.J. Miklavcie, “Theory of Light Propagation incorporating scattering and absorption in turbid media”, Opt. Lett., 30, 792-794, 2005
[10] <http://www.answers.com/topic/principal-components-analysis>, Mathworld, Principal Component Analysis, 2009
[11] < http://mathworld.wolfram.com/LeastSquaresFitting.html>, Mathworld, Least Square Fitting, 2009
[12] <http://www.bibl.liu.se/liupubl/disp/disp97/tek492s.htm>, Dot Gain in Colour Halftones, 2007
137
[13] <http://en.wikipedia.org/wiki/Radial_basis_function>, Wikipedia, Radial Basis Functions, 2009
[14] P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche”, Z. Tech. Phys. 12, 593–601, 1931
[15] Di-Yuan Tzeng, “Spectral Based Color Separation Algorithm Development for Multiple Ink Color Reproduction”, PhD Thesis, Rochester Institute of Technology, 1988
[16] Li Yang, Ink-Paper Interaction, “A Study in Inkjet Color Reproduction”, PhD Thesis, Linkiping University, 2003
[17] Murray, A., “Monochrome Reproduction in Photoengraving”, J. Franklin Inst. 221,721-744, 1936
[18] Neugebauer, H. E. J., “Die Theoretischen Grundlagen des Mehrfarbenbuchdrucks”, (German) Zeitschrift für Wissenshaftliche Photographie Photophysik und Photochemie 36:4, 73-89 (1937) [Reprinted in Proc. SPIE 1184: Neugebauer Memorial Seminar on Color Reproduction, 194-202, 1989
[19] Demichel, M. E., Procédé 26, 17-21, 26-27, 1924
[20] Yule, J. A. C., “Principles of Color Reproduction”, John Wiley & Sons, Inc, 255, 1967
[21] Wyble, D., Berns, RS., “A Critical Review of Spectral Models Applied to Binary Color Printing”, Rochester Institute of Technology, 1999
[22] Yule JAC, Nielsen WJ. “The penetration of light into paper and its effect on halftone reproduction,” Proc. TAGA, 1951
[23] Balasubramanian R. “A printer model for Dot-on-Dot halftone screens.”, Proc SPIE, Color hard copy and graphic arts IV, Vol, 2413, p. 356-364, 1995
[24] Heuberger KJ, Jing ZM, Persiev S. “Color transformations and lookup tables”. TAGA/ISCC Proc; p 863–881, 1992
[25] Kohler. T., Berns, RS., “Reducing metamerism and increasing gamut using five or more colored inks”, Proc. of IS&T Third Technical Symposium on Prepress, proofing and Printing, pg. 24, 1993
[26] Tzeng, DY., Berns, RS., “Spectral-Based Six-Color Separation Minimizing Metamerism”, Proc. CIC, The Eighth IS&T/SID Color Imaging Conference, 2000
[27] DY. Tzeng and R. Berns, “Spectral-Based Ink Selection for Multiple-Ink Printing I. Colorant Estimation of Original Objects”, Proc. IS&T/SID Seventh Color Imaging Conference: Color Science, Systems and Applications, Scottsdale, pg. 106-111, 1998
138
[28] DY. Tzeng and R. Berns, “Spectral-Based Ink Selection for Multiple-Ink Printing II. Optimal Ink Selection”, Proc. IS&T/SID Seventh Color Imaging Conference: Color Science, Systems and Applications, Scottsdale, pg. 182-187, 1999
[29] DY. Tzeng., “Spectral-based color Separation algorithm development for multiple-ink color reproduction”, Ph.D. Dissertation, Rochester Institute of Technology, 1999
[30] David Connah, Ali Alsam, Jon Hardeberg, “Multi-Spectral Imaging: How many sensors do we need”, Proc. IS&T/SID Twelfth Color Imaging Conference, pg 53-58, 2004
[31] H. F. Kaiser, “The varimax criterion for analytical rotation in factor analysis”, Psychometrika, 23, 187, 1958
[32] P. D. Burns, “Analysis of image noise in multispectral color acquisition,” Ph.D. thesis, Center for Imaging Science, Rochester Institute of Technology 1997
[33] F. K¨onig andW. Praefcke, “A multispectral scanner”, in L.W. MacDonald and M. R. Luo, editors, Colour Imaging: Vision and Technology, John Wiley and Sons Ltd, pp. 129-144, 1999
[34] H. Sugiura, T. Kuno, N. Watanabe, N. Matoba, J. Hayashi and Y. Miyake, “Development of high accurate multispectral cameras”, in Proceedings of the International Symposium on Multispectral imaging and Color Reproduction for Digital Archives, Chiba University, Japan, pp. 73-80, 1999
[35] J. K. Eem, H. D. Shin and S. O. Park, “Reconstruction of surface spectral refectances using characteristic vectors of Munsell colors”, in Proceedings of IS&T and SID's 2nd Color Imaging Conference: Color Science, Systems and Applications, Scottsdale, Arizona, pp. 127–31, 1994
[36] L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with a small number of parameters”, Journal of the Optical Society of America - A, 3(10), 1673, 1986
[37] J. Parkkinen, J. Hallikainen and T. Jaaskelainen, “Characteristic spectra of Munsell colors, Journal of the Optical Society of America - A, 6, 318, see http://cs.joensuu.fi/spectral, 1989
[38] W. Wang, M. Hauta-Kasari and S. Toyooka, “Optimal lters-design for measuring colors using unsupervised neural network”, in Proceedings of the 8th Congress of the International Colour Association, AIC Color 97, Kyoto, Japan, vol. I, pp. 419-422, 1997
[39] J. Y. Hardeberg, “On the spectral dimensionality of object colors”, in Proceedings of CGIV'2002, First European Conference on Colour in Graphics, Imaging, and Vision, Poitiers, France, pp. 480-485, 2002
139
[40] A. Johnson and D. W. Wichern, “Applied Multivariate Statistical Analysis”, 3rd Ed. Prentice Hall, New York, 459-486, 1992
[41] J. L. Simonds, “Application of Characteristic Vector Analysis to Photographic and Optical Response Data”, Journal of Optical Society of America. 53, No. 8, 968-974, 1963
[42] D. B. Judd, D. L. MacAdam, and G. Wyszecki, “Spectral Distribution of Typical Daylight as a Function of Correlated Color Temperature”, Journal of Optical Society of America, 54, No. 8, 1031- 1040, 1964
[43] N. Ohta, “Estimating Absorption Bands of Component Dyes by Means of Principal Component Analysis”, Analytical Chemistry, 45, 553-557, 1973
[44] J. Morovic, M. Lou, "The fundamental of Gamut Mapping: A Survey", Journal of Imaging Science and Technology, vol. 45, no. 3, 2001
[45] McDonald, L.W., “Gamut Mapping in Perceptual Color Space”, Proceeding 1st IS&T/SID Color Imaging Conference. Springfield, VA, 1993, 193-196
[46] R. S. Gentile, E. Walowitt and J. P. Allebach, “A comparison of techniques for color gamut mismatch compensation”, Journal of Imaging Technology, vol. 16, pp. 176-181, 1990
[47] E. D. Montag and M. D. Fairchild , “Gamut mapping: Evaluation of chroma clipping techniques for three destination gamuts.” IS&T/SID Sixth Colour Imaging Conference, Scottsdale, 1998, p. 57-61
[48] Mitchell Rosen, N. Ohta, “Spectral Color Processing using an Interim Connection Space”, IS&T/SID Twelfth Colour Imaging Conference, Scottsdale, 2004, p. 187-192
[49] Mitchell Rosen, F. Imai, X. Jiang, N. Ohta, “Spectral Reproduction from Scene to Hardcopy II: Image Processing”, Proceedings of SPIE, the International Society for Optical Engineering, 33-41, 2001
[50] L. Taplin, R. Berns, “Spectral Color Reproduction Based on a Six Color Inkjet Output System”, Proc. IS&T/SID Ninth Color Imaging Conference, 209-213, 2001
[51] DY. Tzeng, R. Berns, “Spectral Based Six Color Separation Minimizing Metamerism”, Proc. IS&T/SID Eighth Color Imaging Conference, pg 342-347, 2000
[52] Y. Chen, R. Berns, L. Taplin, F. Imai, “A Multi-Ink Color Separation Algorithm Maximizing Color Constancy”, Proceeding of IS&T/SID Eleventh Color Imaging Conference, 277-281, 2003
[53] P. Urban, R. Grigat, “Spectral Based Color Separation Using Linear Regression Iteration”, Wiley Periodical, Vol. 31, No. 3, 229-238, 2006
140
[54] P. Urban, M. Rosen, R. Berns, “Fast Spectral-Based Separation of Multispectral Images”, Proceeding of IS&T/SID Fifteenth Color Imaging Conference, 2007
[55] A. Bakke, Ivar Farup, J. Hardeberg, ”Multi-Spectral Gamut Mapping and Visualization – A first attempt”, SPIE, Color imaging X : processing, hardcopy, and applications, Vol 5667, 193-200, 2005
[56] H. Haneishi and Y. Sakuda, “Representing Gamut of Spectral Reflectance by a Polyhedron in High Dimensional Space,” in Proceedings of the Third International Conference on Multispectral Color Science (MCS’01), pp. 5–8, 2001
[57] M. Derhak, M. Rosen, “Spectral Colorimetry Using LabPQR – An Interim Connection Space”, Proceeding of IS&T/SID Twelfth Color Imaging Conference, 246-250, 2004
[58] S. Tsutsumi, M. Rosen, R. Berns, “Spectral Reproduction Using LabPQR: Inverting the Fractional-Area-Coverage-to-Spectra Relationship”, Proceeding of IS&T/SID Fourteenth Color Imaging Conference, 107-110, 2006
[59] S. Tsutsumi, M. Rosen, R. Berns, “Spectral Color Reproduction Using an Interim Connection Space-Based Lookup Table”, Proceeding of IS&T/SID Fifteenth Color Imaging Conference, 184-189, 2007
[60] B. Bastani, B. Funt, J. Dicarlo, “Spectral Reproduction – How many primaries are needed?”, Proceeding of NIP23, 23rd International Conference on Digital Printing Technologies and Digital Fabrication, Anchorage, Alaska, 410-413, 2007
[61] Xiong, W., Shi, L., Funt, B., "Illumination Estimation via Thin-Plate Spline Interpolation", Proceeding of IS&T/SID Fifteenth Color Imaging Conference, 2007
[62] J. B. Tenenbaum, V. de Silva, J. C. Langford (2000). "A global geometric framework for nonlinear dimensionality reduction", Science 290 (5500): 2319-2323, 22, 2000
[63] G.A.F. Seber, Multivariate Observations”, Wiley, 1984
[64] B. Bastani, B. Cressman, B. Funt, "Calibrated Color Mapping Between LCD and CRT Displays: A Case Study", Color Research and Application, Volume 30, Issue 6, Date: December 2005, Pages: 438-447
[65] Barnard, K. Martin, L., Funt, B.V. and Coath, A., "A Data Set for Color Research", Color Research and Application, vol. 27, no. 3, pp. 140-147, 2002. (Data from: www.cs.sfu.ca/~colour )
[66] J. Ferguson , P. A. Staley, “Least squares piecewise cubic curve fitting”, Communications of the ACM, Volume 16 , Issue 6, 1973
141
[67] G. Finlayson, S. Hordley and I. Tastl, "Gamut Constrained Illuminant Estimation”, International Journal of Computer Vision, vol. 67 , no. 1, 2006
[68] B. Bastani, B. Cressman, M. Shaw, “Sparse Cellular Neugebauer Model for N-ink Printers,” Proceeding of IS&T/SID Fourth Color Imaging Conference: Color Science, Systems and Applications, Scottsdale, pp. 58-60, 1996
[69] N. Katoh and M. Ito, “Applying Non-linear Compression to the Three-dimensional Gamut Mapping”, The Journal of Imaging Science and Technology, vol. 44, no. 4, pp. 328-333, 1999
[70] T. Cholewo and S. Love, “Gamut Boundary Determination Using Alpha-Shapes,” Proceeding of IS&T/SID Seventh Color Imaging Conference: Color Science, Systems and Applications, Scottsdale, pp. 200-204, 1999
[71] C. McIntosh and G. Hamarneh “Is a Single Energy Functional Sufficient? Adaptive Energy Functionals and Automatic Initialization”, In Lecture Notes in Computer Science, Medical Image Computing and Computer-Assisted Intervention (MICCAI), pp. 503-510, 2007
[72] Z. Ma, Y. Bi, Inventor; Hewlett-Packard Company, “Inkjet Inks Having Reduced Bronzing”, International Patent PCT/US2006/017315, 2005
[73] L. Tsang, J. Moffatt, M. Austin, Inventor; Hewlett-Packard Company, “Additives to eliminate bronzing of inkjet ink formulations on specialty quick-dry inkjet photographic media”, United States Patent US 7052535, 2006
[74] < http://www.aviangroupusa.com/MCRL_GCMS.php> , Murakami GonioSpectrophotometer, 2008
[75] Li Yang, “Light Media Interaction in print color reproduction”, Proc. NIP22, 22nd International Conference on Digital Printing Technologies and Digital Fabrication,, Tutorial, 2006
[76] Bastani, B, Funt, B, Dicarlo, J, “Spectral Reproduction- How Many Primaries Are Needed?”, Proc. NIP23, 23rd International Conference on Digital Printing Technologies and Digital Fabrication, Alaska, p. 410-413, 2007
[77] Bastani, B, Funt, B, “Spectral Gamut Mapping and Gamut Concavity”, Proceeding of IS&T/SID Fourteenth Color Imaging Conference: Color Science, Systems and Applications, Albuquerque, NM, p. 218-221, 2007
[78] Bastani, B., Funt, B., “Spectral Modeling of an n-Ink Printer via Thin Plate Spline Interpolation”, Proc. NIP24, 24th International Conference on Digital Printing Technologies and Digital Fabrication, 2008
[79] Bastani, B., Funt, B., “Geodesic Based Ink Separation for Spectral Printing”, Proceeding of IS&T/SID Sixteenth Color Imaging Conference: Color Science, Systems and Applications, 2008
142
[80] <http://www.mathreference.com/ca-int,simp.html>,Mathreference, The Volume of a Simplex, 2008
[81] Sharma G, editor, “Digital Color Imaging Handbook”, CRC Press, Boca Ranton, FL, 2003
[82] Li, X., Li, CJ, Luo, M. R., Pointer, M., Cho, M. and Kim, J., “A New Colour Gamut for Object Colours”, Fifth Colour Imaging Conference, pages 283-287, 2007
[83] Han J, Kamber M, "Data Mining: Concepts and techniques", Chapter 8, pp335-485, Morgan Kaufmann, 2001
[84] < http://www.cis.rit.edu/mcsl/online/cie.php >, RIT Useful Color Data, 2009
[85] B. Bastani, B. Funt, “Spectral Gamut Characteristics Based on Number of Primaries and Their Characteristics”, Proc. NIP25, 25rd International Conference on Digital Printing Technologies and Digital Fabrication, Louisville, Kentucky, 410-413, 2007
[86] Joshua B. Tenenbaum, Vin de Silva, and John C. Langford, “A Global Geometric Framework for Nonlinear Dimensionality Reduction”, Science 22 2319-2323, 200
[87] R. Balasubramanian, “Optimization of the spectral Neugebauer model for printer characterization”, Journal of Electronic Imaging, Vol. 8, 156, 1999
[88] < http://mathworld.wolfram.com/DijkstrasAlgorithm.html>, Wolfram Mathworld, 2009
[89] F. Nakaya and N. Ohta, “Spectral encoding / decoding using LabRGB”, Fifteenth Colour Imaging Conference, 2007
[90] F. Nakaya and N. Ohta, “Applying LabRGB to Real Multi-Spectral Images”. , Sixteenth Colour Imaging Conference, 2008
[91] <http://en.wikipedia.org/wiki/Thin_plate_spline>, Wikipedia, Thin Plate Spline, 2009
[92] < http://www.mathworks.com/support/tech-notes/1500/1508.html>, Matlab, Curve Fitting